Story Idea:
Lord Potter and the Philosopher's Stone (first in series?)
Summary: Lord Potter (from an alternate universe where Riddle never became Voldemort) is transferred (similar to a Quantum Leap transfer) to his 11 (or 12 if he went through one year of Hogwarts) year old self in the cannon universe.
Differences between universes (explored at the beginning of each chapter through quotes from Ariana's biography or something similar): The prime difference is that Ariana Dumbledore tripped during the duel between her older brothers and Grindelwald. The curse that in cannon hit her instead hit Albus, killing him. This triggered another burst of magic from Ariana which killed Grindelwald, knocked out Aberforth, and alerted the authorities. (Grindelwald's death caused the second world war to not invade the wizarding world as much.)
She got the help she needed to control her magic (led by Healer Longbottom) and become a very powerful witch. The research into the help she needs also helped squibs expand their crippled magical cores as long as they are caught young enough. However, even the older ones can get some magic, just nowhere near as strong as the ones caught younger.
After being healed, she took a position at Hogwarts, where she was put in charge of new student notifications. She looked in the book/scroll/whatever they used in that universe and noticed that certain children were being abused because of their magic, and there were some that were placed in muggle orphanages. Knowing how bad muggles can treat young magical folk, she fights for magical orphans/abuse victims to get adopted/fostered by magical families (or at the very least a magical orphanage) that help the children.
(One of her first cases was a young Tom Riddle, who receiving the love and care he needed, never became Voldemort. Instead, he took his classes and became one of the school's finest DADA teachers. He told the staff about the Chamber of Secrets, so the basilisk was taken care of years ago.)
In addition, she made it so that when a muggleborn child in a good relationship with their parents displays their first accidental magic, a response team gets sent out immediately to not only help keep secrecy, but to slowly integrate the family into the magical world. When she became headmistress through her works, she made sure that she and her staff enforced rules fairly and evenly, even if sometimes a child had to be suspended (making Slytherin House a respected house instead of a dumping ground for dark wizards/witches). The Hogwarts house elves made sure that any dark items brought in were reported to the staff, who then confiscated them. These kinds of actions have made her one of the most loved headmistresses of Hogwarts.
Because there was no Grindelwald, no Voldemort, and young muggleborns aren't seen as outsiders, blood status prejudices, although still around, don't matter much except for some minor, stuck-up pureblood families. Wannabe Dark Lords are quickly swatted down before they can do anything.
And because there isn't a Dark Lord, there is no prophecy.
Some character differences:
Fenrir wasn't supported by an evil man and was therefore captured before attacking Remus Lupin.
Severus Snape was removed from his abusive household and put with a loving family.
When Petunia Evans begged to be allowed to go to Hogwarts, she was tested and found to be a squib. After her treatment, she was admitted.
The Marauders did come close to crossing the line with a few of their pranks, but nothing really mean spirited ever came to pass.
Snape and Lily did try dating, but decided they would be better friends. He's still a very good family friend.
Lily ended up with James.
Other possible pairings: Sirius ended up with Petunia, Pettigrew ended up with Bellatrix, Lupin ended up with (?).
Since there was no war, Harry's parents lived, Neville's parents are okay, etc. Harry grew up in a loving environment, had siblings, etc. Because of his connection to the Longbottoms (Alice is his godmother), he is also connected to Headmistress Ariana (who kept close ties with the Longbottoms due to her healing). Through her and (especially) Aberforth (who teaches History of Magic), he has learned what a jerk that Albus was.
No war = more students = more classes & teachers. A good headmistress not trying to run a secret war and get involved in every single little political thing and actually cares for her students = Hogwarts being an actually good school. Peeves has been banished to a certain area (he's used in upper level DADA classes). The Chamber of Secrets is open to the students to examine. Etc.
Series 1 Idea: Knowledgeable Potter
Lord Harry Potter (around, say, 30-45 years old) from that universe swaps places with cannon Harry Potter. He has more knowledge than cannon Harry (such as Potter holdings, spells and the like), doesn't know other things (the prophecy, Voldemort), and gets some things wrong (Petunia is his loving aunt married to Sirius Black). He has to deal with everything (preferably through the entire series) with a man (Dumbledore) he doesn't trust, going to a poor shadow of his Hogwarts. Since he is in an abused 11/12 year old body, he doesn't have the power to do all of the spells he knows.
In addition, there could be interludes where cannon Potter and Lord Potter invade each other's dreams. This could be used to pass information to the reader, letting them know about the above changes (in addition to Ariana's biography clips).
Series Idea 1A: Loved Potter
While the above series is taking place, we follow cannon Potter through the other universe, meeting family, friends, etc. His adult body holding his 11/12 year old mind would be transformed into a child's body so he can take the classes he would need at his age level.
Series 2 Idea: Powerful Potter
Instead of Lord Potter switching, cannon Potter gets pulled into LP's universe (probably by LP) and sees what wizarding Britain could be. He is trained by the Hogwarts school and sent back to his universe to works towards a better society.