Hermione Granger
Minerva McGonagall
During the Final Battle, Bellatrix hits Hermione with a dark spell that de-ages her and sends her back in time to the same year that Minerva McGonagall was born.
As Bellatrix finished her final wand movement she whispered the incantation and poured all of her magic into it. Unlike some spells that get more powerful the louder you say them, this one got more powerful the more softly you said it, which is why Bellatrix whispered the incantation. A purple-green jet of light flew straight at Hermione who tried to duck but it followed her. As it hit her, it reacted with her magic and the magic from her time turner.
Bellatrix, being blinded by her prejudices and insanity, used a spell that was designed to affect muggles, as they have no magic. Well, they have no usable magic; life itself is part of magic. However, Hermione was a witch; a very powerful witch; a very powerful witch who was, in fact, wearing a time turner.
And so the spell reacted with her magic and the magic of the time turner in a unique and interesting way, instead of causing her to age and turn into an old crone barely able to stand, it caused her to age in reverse. As far as the time turner's magic, it sucked her back in time.
When she arrived in Hogwarts, it was the same year that Minerva McGonagall was born. Well, that is to say the McGonagall Family had an infant girl who was stillborn and Hermione Granger arrived as an unknown infant. So Hermione Granger arrived in 1935, as a baby, only a week old.
So, Headmaster Dippet gave her over to the McGonagall family who immediately blood adopted her. After all, it was his younger sister, Isobel, who had had the still born baby. Being an infant, she became a full McGonagall in blood and genetics, retaining a little of her own previous DNA. They named her Minerva Athena McGonagall, because she arrived as an unknown week old baby, the magic of Hogwarts transporting her to the hospital wing.
Now it is important to say that Hermione, when she de-aged and became Minerva, did not retain all of her memories, but she did retain some impressions in a gut feeling. They weren't memories; they were gut feelings. She had a gut feeling that the Marauders were behind most of the pranks, even without proof; later this would extend to Fred and George Weasley. She had a gut feeling that Harry would not be happy with the Muggles, that he could be a great seeker, and that he was a good boy. She had a gut feeling that Hermione could be trusted with the time turner in her third year.
The rest is History.
AN: If you look at the Harry Potter wiki at Minerva's biography, there are a few things that are similar. First, Minerva and Hermione both have a love of cats. Minerva could have gone to Ravenclaw but went to Gryffindor instead; same with Hermione. Hermione has an interest in transfiguration and a talent for it; so did Minerva. Minerva is a strict disciplinarian; if Hermione became a teacher, she would be too. Both have a love of books and both are logical. So here you go.
Again Credit Goes to IrishDanceringRules - for beta reading and to Jostanos for providing the title.