That would actually be close to Voldemort's stay in Hogwarts. Voldy, being a poor orphan, had to prove himself against an entire house of prejudiced, uppity, middle-class dicks.So you've got this kid who's being bullied and all of a sudden he has an escape from that and as the sorting hat says "wants to prove himself". You can give him a lot of depth from that. Make him an academic tryhard and that's how he gets in with Hermione, make him really sporty and that's how he gets in with the Weasleys, and make him not having any classist pretensions a source of conflict since a solid chunk of Slytherin house is upper middle class. Have Harry be the aspirant side of ambition and cunning, as opposed to "normal" Slytherins who represent the upper crust's hold on wealth and power and their grooming their kids to inherit it.
We could have had Voldy and Harry start out as ambition-cunning aspirants who came from similar backgrounds and struggled in proving themselves talented wizards but ultimately took very different paths. In this head-cannon, I imagine several factors in play. One is that wizards in Voldy's day were far more prejudiced against muggles than during Harry's day. Combine that with Voldy discovering his father was a "filthy" muggle, and that sets him down a path of darkness. Harry has a "mudblood" for a mother, but at least his mother was magical, and while Slytherins are obviously going to be dicks to Harry about it, at least it won't be as bad.
Second, Harry could have ended up with supportive friends while Voldy only ended up with sycophants leeching his talent. This could be because 1) Harry tries harder to reach across house boundaries, especially the prejudice against Slytherins as bad guys, and 2) the house boundaries are a bit more relaxed than in Voldy's day, making it possible for Harry to succeed in this endeavor. Harry earns the friendship and love of Hermione and Ron through his hard work in sports and academics. Harry has rough patches with Draco at first, but they become frenemies with a certain mutual understanding. Thus, when the Deatheaters tempt Harry to join the dark side, Harry refuses because he has a healthier sense of self and a support group.
tl;dr The Sorting Hat was right; Harry would have done well in Slytherin.