Harry and Dudley raised as Petunia's twin sons
Petunia Dursley is in an abusive marriage with Vernon Dursley. So far, he has only shown love and affection towards their son, Dudley, but she is scared of that changing. From the moment she is pregnant with Dudley, she starts preparing to leave him (eg saving money, thinking about where to go and making plans). When Harry Potter is dropped on her doorstep, she is devastated at the loss of her sister and determined to care for Harry. However, Vernon disregards her loss as Lily was always a 'freak' and puts the one year old who just lost both his parents in the cupboard under the stairs. Petunia makes a decision that night, and (maybe after some more time preparing) she takes Dudley and Harry and leaves. She starts over with new names for all of them, and begins a new life as a single mother of twin boys (maybe a new birthday in between their birthdays?). She treats them equally and tells them about their aunt Lily who was a witch and died during a wizarding war. Maybe she tells Harry and Dudley that Harry is Lily's son, but to the outside world they are twins.
I think she will need a witch/wizard/squib/goblin to help her, but I want Harry to be completely unrecognizable to the wizarding world. So maybe someone points Petunia to some books on rituals/potions/spells this witch/wizard/squib/goblin can help her perform to make sure Harry's scar is hidden and his looks change a bit.
When Harry starts showing signs of magic, Petunia starts learning more about the wizarding world to best prepare Harry when he will inevitably go Hogwarts. This includes going to Diagon Alley and a bit normalization of magic for all three. When Harry turns 11 (on their fake birthday) they go to Diagon with the Hogwarts teacher in charge of accompanying muggleborns. When he arrives to Hogwarts, no one suspects he is Harry Potter, and he can start his education as a normal wizard, while the rest of the wizarding world is searching for the Boy-Who-Lived. Albus won't manipulate him into any shenanigans like the philosopher's stone. Harry is writing home frequently to his aunt/mother and Dudley, who can write back because they have an owl, or direct it with muggle post to Hogwarts.