Whenever the Hat is considering more than one house, I think you can have some influence on it ... if the traits you truly value more match with the house you think you belong in. But I do also think that the potential that the Hat can see also has quite a bit of significance in the final decision that is made.

In the example of Harry, I felt he exhibited Gryffindor traits in that moment without second thought. He definitely the nerve to deny the Hat's consideration of Slytherin because he dubbed those in Slytherin as the "wrong sort" after not being impressed at all with Draco and not wanting to be in the same house Voldemort and other "bad" witches and wizards (plus I'm sure he wanted to be in the same house as his newfound BFF Ron). But it should be noted that he displayed no interest and not a single ounce of ambition or cunning when the Hat said Slytherin would help him on the way to greatness, which is probably what made the Hat go with Gryffindor in the end.

Hermione is an interesting sort, and I can totally see why the Hat spend a bit of time between Gryffindor and Ravenclaw. Instantly it probably considered Ravenclaw as she is incredibly intelligent and values knowledge, but this statement made by Hermione in PS/SS on the train helps solidify the Gryffindor sort: "... I hope I'm in Gryffindor, it sounds by far the best; I hear Dumbledore himself was in it, but I suppose Ravenclaw wouldn't be too bad." This, to me, shows that she values the traits of Gryffindor a bit more than that of Ravenclaw. This is again confirmed later in the book when she tells Harry "Books! And cleverness! There are more important things -- friendship and bravery..." (I can see McGonagall and Flitwick's sortings going a similar way -- the Hat made it's final decision based on the traits each individual valued more.)

In the case of Albus Potter, he wound up in Slytherin even though he didn't want to be there. I do wish the play-that-shall-not-be-named had gone a little more in depth with his sorting to better show us how the final decision came to be. I'm not questioning the Slytherin placement though as I assumed Albus would be sorted there before it was revealed (I was on board with the theory all those years ago that assumed his initials, ASP, was a hint to where he'd be sorted) and can definitely see that his wants/desires/values tend to align more with Slytherin (his interest in his Harry's fame and past, the need to prove himself, his determination to distance himself from Harry's shadow, etc.).

As for Neville, the fact that he was arguing with the Hat "for a long time" about where he belonged ... gives me Gryffindor vibes for some reason? Like, he's determined to challenge the Hat's decision rather than accepting it like a Hufflepuff probably would? It's hard to say as we don't know word-for-word how that conversation went. But, in this instance, I would agree that the Hat saw potential in Neville as a Gryffindor and knows that he values those traits even though he doesn't think he exhibits those traits himself. (Maybe this is the case with Peter Pettigrew; the Hat saw potential and knew that young Peter valued bravery but, unlike Neville, ended up not living up to that potential.)

Gilderoy Lockhart, though -- Sorting Hat, what were you thinking?! Lockhart's Ravenclaw/Slytherin Hatstall is something that I am incredibly fascinated by. I want to know how he "narrowly escaped into Ravenclaw" when it seems that his ambition and desire for fame, popularity, being the best, etc. was always his priority. Maybe this is just a case of the Hat seeing more potential in Ravenclaw than Slytherin for him and that it could influence him not be so self-centred?

Thank you for coming to my TED Talk.