I go by the books, which I only have copies of, because I am not Rowling.
Consider the lists at the bottom prompts. Challenge...
What's in a Name?
Generally speaking, when something doesn't really make sense in fantasy books or games or movies, I like to just call 'magic' on it. It's called suspension of disbelief and some people are better at it than others. I'm usually pretty good... good, that is, until I decide I want to write something about that world. My Hyrulian history is a testament to this: consistent writing usually requires references so as to not contradict yourself because calling 'magic' is so easy.
So, name magic.
The power found in one's true name is an idea common to many cultures. Socrates wrote about it. The ancient Egyptians used theirs in rituals and guarded them jealously: to lose one's name was to be denied an afterlife and wander the desert eternally. Unbaptised and thus unnamed children were thought to be at a risk of being stolen by the fairies. These are only a few examples but names are often considered sacred.
So what might that mean for a school of magic named after a pig with a skin disease?
To be fair, a pig or boar totem often implies determination (stubborn as a pig), fertility and abundance (some gods ritually at pork to maintain immortality), wealth, prosperity and luck (oft will pigs in suits appear in editorials and cartoons), and cleanliness and organization (which amuses me greatly, but apparently refers to self reliance).
Hogwarts lasted over a millennium and at a time looked after its village, Hogsmead – stubborn and fertile/prosperous. It is also a standalone enterprise: private school, grows its own crops, little communication with the outside world during much of the year. It is ambiguously organized (rearranges itself, poor communication... but it makes sense to wizards, maybe?).
This is all well and good, but the warty bits? Hogwarts has many faults, much touched upon by other authors, like segregation and the DADA curse, so I'll simply ask: why would someone deliberately give such a name to a school? Give it good vibes but off somehow? There is a rumour (on Pottermore?) about rumours of Rowena Ravenclaw having a dream about a warty hog leading her to the castle grounds, but that sounds like a myth that pops up later to explain something strange.
Instead consider what would Hogwarts be if it was not 'Hogwarts?'
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'Hogwarts' is modern English. The school was founded around the end of the tenth century or beginning of the eleventh, in Scotland, possibly in the Banffshire region of Moray. This is a time and region where the languages of the Picts had been spoken and were just becoming what we would know as old Gaelic.
At the same period in Britain the other languages spoken were Old English/Anglo-Saxon (Mercian, Kentish, West Saxon, Northumbian dialects), Old Welsh, Old Norse, and (in high and religious classes only) corrupted Latin. All of these languages varied from region to region and period to period, sometimes vastly, since most could not read or write and only learned by ear. Even if literacy was higher the transmission of text by scribing is imprecise since the languages did not at this time have a consistent writing system, set of spellings, and texts would contain scribal errors. The printing press helped regulate language, but was only invented in Germany in 1450. Lots of room for changes.
Though 'Hogwart' in Old English is still 'Hoggwearte' or 'Hogcwearte,' but that doesn't change the question of why use such names when they leave the same bloody problem.
Which leaves the alternatives, and from them another question: who named the castle? There are three options.
The first follows from the logic that the Sorting Hat says that the Founders built the school, not the castle. It's implied that they built the building too, but as that is not stated there is fiddling room. The Founders may have just expanded. So, if the castle was acquired, it was probably already named – most likely in Pict or another earlier Celtic tongue, and the name morphed over time.
Alternately the Founders named the castle, in one of two scenarios: naming as original builders or renaming as acquisitors. It is doubtful the Founders would have found reason to rename the place, especially if they bought or were gifted with the original grounds as a bequeathed as payment for a deed or the good will of the local king or some such. If the original name was lost (abandonment) or changed (which usually happens for political reasons, or because the old name was considered unlucky) the Founders might be credited but this is less likely. Here to Hogwarts is the end of a morphological change.
The last possibility is 'Hogwarts' (or Hogcweart) was a later renaming of the castle, for some political reason, or even as a curse, if we apply name magic. After all, the Hogwarts we know has some serious issues under its skin.
Since Option One can be labelled a Goidelic name, and Three isn't worth looking at since if 'Hogwarts' is a later change the original could be anything – I pass the buck to other writers. So we skip to Two.
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The founders are of rather diverse origins. Being magical and founding a school in Scotland, they all likely speak some Gaelic language(s) and Latin.
Gryffindor, himself from south-western England, has an Anglo-Saxon name, which is to be expected, and a French or Latin surname, which is less so, as the Normans don't arrive until 1066, bringing more Latin based words although many Germanic tribes were highly Romanized towards the end of the Western empire. Godric means 'he who rules with God' or 'he who rules well.' Gryffindor is the literal French 'golden griffin,' le griffon d'or. In Latin it would be gryps/gryphus de auro. So, leave Griffindor, or chose Gryphus d'auro or Gryphondor? I, at least, am partial to a more Latinised name.
Godric Gryphondor speaks Old English and may use a Frankish dialect.
Hufflepuff, despite being identified as Welsh, has a Norse personal name and an Old English or Norse surname. Before I wiki'd her I thought she was from the upper east coast of Britain. Helga is from Helge from heilagr, which means brave or prosperous. 'Hufflepuff' could be set of onomatopoeia (huff and puff), a Norweigan expression of disgust or horror (huff, and how would her family have earned that one I wonder?), or the Old English pyffian which just means 'puff' according to wiktionary, or Old English for 'pastry' ('puff' again).
I am tempted to call her Helge Hwilpyff. (ff-w-v-b is a common sound change. Hwil is also Old English for 'while' and related to Old Norse hvila 'to rest.') But the Hufflpyff clan might just be long-winded or something similar. She may speak either Old Welsh (but probably not) or Old Norse or Old English.
As a minor aside: on the subject of huff as horror, if names have power, why do you see families called 'Weasley' and 'Malfoy'? My personal theory is that magical clans can earn monikers. At some point in the past both families had someone (or several someones) who lived up to such a name. As the Malfoys continue to act in 'bad faith,' so the name acts as a warning.
Slytherin is rather interesting, as he may hail from the east coast ('from fen'), and his surname is English, but his personal name is the same as the surname of the Iberian noble family. And wouldn't it be curious if his name had gotten reversed? The name Salazar, from the Castilian Sala (hall) and Basque zahar (old), was first recorded in northern Spain in 924 in the Sarasazo denomination, and earlier in Basque, where it originated. Did a branch of a noble family cross the Channel and influence notions of blood purity?
Slytherin, by contrast, is much less interesting. Proto-German slidrianan became Old English slidrian and Middle English slitheren or sliddren: to slide, slither, creep. Middle English is not technically relevant, but we see how the word becomes familiar, and 'Middle English' begins in 1066, officially, which is not too far off. And so we have Slidrian Salazar, an English-born Basque noble, speaking Old English, and Castilian and/or Basque.
Ravenclaw is also odd, as she is said to be 'from glen' which presumably means Scotland, but she is named after either the mythological daughter of the Anglo-Saxon chief Hengist, a fifth century legend, or possibly for horsehair (the Welsh Rhonwen from rhawn) which may still relate to the legend. Alternate spellings are Latinised German/English: Ronwen, Renwein, and Romwenna which mean fame, white lance, or white hair.
Her family name may be Old English hraefn and clawu, raven and claw. Other options are: [old Saxon hraban][Old High German raban] or [Old Norse hrafn] and [Proto-German *klawo] and its derivatives. Whichever is the case: she is not a Scot. Rhonwen Raefnclaw was probably Welsh or a Briton, speaking Old Welsh and/or Old English.
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That done, we now know what languages Hogwarts might be derived from, if it only became corrupted overtime. Old English, likely known in some degree to all founders, is already out of the running. French/Frankish, Basque and Castilian are out since only one founder might know each, and for reasons of discrimination/nationality/xenophobia are unlikely to stick. Latin is a maybe, since sometimes in old texts it seems like the Romans pissed rainbows, or possibly the salt with which they paid soldiers. Latin, at least, would be respected. Gaelic/Pict is the most likely, due to the location. The Welsh language belongs on the island and two Founders may have spoken it, so it is a maybe. It also still resembles (to a point) Gaelic and English at this period, though it was already long since distinct (they split circa 550 CE). Norse is an outside chance: there are Nordic peoples at the northern-most point of Scotland around the Founders time, and Helge may be Norse herself.
So probably Gaelic, or possibly Latin, maybe Welsh and very distant chance of Norse (to be ignored for now).
While some of these are really stretching it all of the following are a few examples I came up with which could have changed into 'Hogwarts.'
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Latin
Unlike Welsh and Gaelic down the page, these should be fairly accurate, what with me being a Classicist. Medieval Latin will contain borrowed words and strange grammar made stranger by being in a universe where Hogwarts is real.
Hagyad (Hagia damma: holy deer.) Namely, the white stag of many legends. If an otherworldly being had led Rhonwen rather than a pig leading Rowenna, that would certainly be auspicious.
Hocaltar (Hoc altaria: the high alter) An altaria would involve sacrifice, so perhaps a shamanic/druidic ritual site preceded the castle.
Hocardem (Holce redemit: A drachma bought it.) I'm taking holce to mean 'a weight of gold' rather than actual drachma. The idea is from the citadel of Carthage, the Bursa, which means bull's hide. Legend held that King Hiarbus allowed Dido to buy as much land as she could cover in a bull's hide: she cut the hide into strips and surrounded a hill with it. Naming your castle for how you bought it has a precedent.
Alternatively: Holco redemimus: We bought (it) with grain. Holco is an ablative of price from holcus: grain, barley. This would look exactly the same as above. Hocardem-Hoguard-Hogwart.
Hocoultab (Hac/Hoc ouls tabem: Here, beyond the melt; Here, far from the plague; Here, far from corruption.) I like the second, for irony (re: warts), and the first which may refer to melt waters (which fill the lake).
Ogcarte (Occare arte: to harrow closely/straight) If built on farmland.
Ogertor(Oggessit/Oggerit torr: the hill was given)Ob-gero in perfect passive, torr is old Gaelic and English. Another acquisition method.
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Welsh
I found an Old Welsh gloss online, and completely disregarded proper syntax. I find this option doubtful, so it is lacking effort.
Haurguird ([h]aur guird: gold green) A descriptor, either of the countryside, or possibly the stone of the castle.
Hocollt (ho coll torr/tir/tol: from the hazel hill/land/tower) Describing the greenery or the masonry.
Huguarth (Huch guarth: sow opposition/resistance) Inside and out.
Ughalt (Uch altaur: above the alter) Another reference to an older site.
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Gaelic.
Not having the necessary texts for early Gaelic or late Pict, I just spent some quality time with MacFarlane's Scottish Gaelic dictionary, and pretended that it wasn't published a century ago. So still fudging it since this obviously isn't period and I don't know the modern grammar, but it's an interesting exercise, anyway.
I spent more time on this since Gaelic seems more likely to me, but remember that these are modern words glued together: basically what Rowling did, but mangling someone else's language instead of my own. This is also not the complete list, just some interesting ones.
Aghaird: Prosperous Height (Hill) or State
Aghairt: Arthur's Fortune. (Associated with Camelot or a different Arthur, or just a local myth)
Aigairetol: In possession of the watching hill
Aogairtor: Shepherd Tower (Used in the sense of leads by teaching, and protecting the surrounding communities)
Beucallaidh: Bellowing Rage (Formerly a Pict stronghold? Or maybe the wind howls around the cliffs. Beucallaidh-Bheucarait-Heugart)
Bheoiceiltor: A living, concealed tower (A stretch, but I like the idea. Bheoiceiltor-Heogeirt-Hogweart)
Bocainoillt: Goblin's Terror (A stretch, and not really feasible, but... Bocainoillt-Bhogaillt-Hogwairt.)
Boghaurd: Bow and Hammer (Past battle site)
Eagearrtoll: Marked Hilltop (A cairn, foundation ruins, a Pict double disk and z-rod engraving, Ogham...)
Fagairt: Forsaken by Arthur. (Associated with Camelot or an ally unaided.)
Fagghallda: Left by Foreigners (Former settlement: the name given by locals when the Founders arrive. Fagghallda-Phagualt-Haguart)
Frithguoillt: Very frightful forest (A stretch, but appropriate. Frithguoillt Hitguort Hogwart)
Poigallt: Kisses the stream (Shores of Black Lake)
Poigather: Kisses the Sky (lofty aspirations, tall towers...)
Uaighairt: Arthur's Grave. (The Black Lake and Avalon or a cairn?)
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So what would you name Hogwarts?