After much thought the hat decided that while Dobby had called it Mr, that wasn't right. The hat decided it had much more in common with the various females it had seen and been exposed to through the centuries but that still left the matter of a name. For more than a day the hat thought through the many students she had sat on the head of, all of their names. None of them really appealed to the hat though, she wanted a name that was all her own.
It occurred to the hat that it might be a good idea for her to come up with a name that went with her job. She was the official sorter of students, maybe there was a way to signify that with her name however as she couldn't read and had no way of turning pages even if she could read there was no way to use a baby name book to find names that meant to sort or classify. The hat had never heard of the internet but that wouldn't have helped within the magical confines of Hogwarts anyway. Finally she recalled a former Headmistress that had been the Muggle Studies teacher. That Head had been particularly interested in the muggle study of science and she would often read aloud muggle books or cast a spell that allowed them to read themselves aloud to her as she worked on other things. Muggles classified animals into different categories and those categories had names, if only she could remember the names maybe she could come up with her name. She wracked her, well whatever it was that passed for her brain, for an hour before they came to her: domain, kingdom, phylum, class, order, family, genus, species. Somewhere in there was her name. Most of the words would seem odd as names however she knew she remembered students will names similar to genus and phylum. Hmmm, phylum, phylum, phyla, Phyla...Phyla what? Phyla Jean! That's it! "I am Phyla Jean Hat!" She pronounced aloud.
She must have spoken louder than she meant to because a response came from the portrait of Phinneas Nigellus Black, "For what possible reason would a hat need a name?"
While the hat formulated an answer, Armando Dippet replied, "Oh hush Phinneas, the poor hat has been without a name for its entire life, why wouldn't it want one?"
"I still fail to see why it would need a name, and it is not alive," Black retorted.
"Not alive?" Dilys Dirwent jumped into the conversation, "Then what is it if not alive?"
"It is merely an object bespelled by the founders," Black drawled, met with cheers of "hear hear" from a few other portraits.
An argument began between the many portraits as to whether the hat was alive or not and whether it needed a name. The hat sat dazed, she listened to the volley of the conversational ball, finally hearing herself referred to as an it one too many times she shouted, "I AM NOT AN IT!"
The room went quiet, all those that could turned to look at her.
"What did you say my dear hat?" inquired Dippet.
"I am not an it. My name is Phyla Jean Hat, and I am a female! You don't ever have to talk to me again, most of you never would anyway, and few if any of you ever have, but I am self-aware and capable of thought. I am aware that this does not meet the definition of alive by most standards but I fail to see how else you could classify me. So no matter what you think I know who I am and I am not seeking either your permission or your approval to name myself. I am simply saying that is who I am!"
The room went quiet, the portraits exchanged glances, but no one seemed to know what to say. The voice that broke the silence shocked them all.
"Hello Phyla Jean I am pleased to make your acquaintance."
"Thank you Professor Black."
Several others excited the sentiment then the office settled once again into silence buy this time for PJ it was a satisfying quiet. She was now ready for Dobby to return so that she could properly introduce herself to the elf. The thought also occurred to her that she should probably properly introduce herself to Dumbledore and McGonagall at the earliest possible opportunity. Perhaps she might also mention that they should look after Winky.