Disclaimer: All characters and objects belong to J. K. Rowling. I am not attempting to use them for profit.
This idea has been kicking around in my head for ages. Neville really is a true Gryffindor, as evidenced by the Hat's reaction to him in Book 7. So, why did the hat take 'a full minute' to decide where to put him? Well, it didn't...
"Hmmmm…" said the little voice in his ear, "Honest, loyal…good mind, but you don't rely on it enough to be a Ravenclaw…very little ambition…courage? Oh, dear me yes, you have courage. It'll be Gryffindor, then."
No, I'm not, thought Neville Longbottom, I'm useless and stupid. Everyone says so. Just make me a Hufflepuff.
"My dear boy," said the voice, "If you were useless, you wouldn't belong in Hufflepuff. As it happens, you would do better in Gryffindor, but Hufflepuff, I'll have you know, is not as useless as everyone seems to think."
I'm not a Gryffindor, thought Neville, Gryffindors are the brave, talented ones. Gran always says I'm not half the wizard my father was.
"That, my boy," said the Hat, "Is because your father was a grown man when you were born. As a boy, he was not so very different from you."
That's not true! My father was brave, and Gran says I'm useless! He wasn't useless! He wasn't!
"Neither are you, boy, no matter how intent you are on denying it."
But I am!
"I've sorted a hundred thousand students in my day, child," said the Hat, "I've never been wrong. A few times, students have chosen their own way and gone wrong, but I have always seen truth."
"Trust me," said the Hat soothingly, "I know my business. You don't know yourself after eleven years as well as I know you from thirty seconds. I know where you belong, and it's not where you think."
Neville could hear the buzz of conversation starting up in the hall as the upperclassmen grew bored with the speed at which events were developing.
Please, just sort me, he thought, Stop making fun of me and let me go. You know where I actually go, so stop making stupid jokes and put me there.
"All right, then," said the hat. There was a long pause as the hat shifted and seemed to consider.
"No," said the hat, finally, "You gave me the choice, and I'll stand by it. GRYFFINDOR!"
Neville jumped off and started to run with the hat still on. As he ran back, flushing a crimson the shade of his new house's banner, he just heard the hat say, "You'll thank me in time, lad."
He did. Eventually.