A/N: This one is a bit different. This is more of a missing moment than something that didn't happen that should have. This is something that is known to have happened but that is not shown in the books: Neville and the Sorting Hat arguing over which house he should go into. This wasn't actually shown in the canon HP books but:

From the book Hogwarts: An Incomplete and Unreliable Guide: In Neville's case, the Hat was determined to place him in Gryffindor: Neville, intimidated by that house's reputation for bravery, requested a placing in Hufflepuff. Their silent wrangling resulted in triumph for the Hat. This is how U think it might have gone down.

...

"Longbottom, Neville."

Neville fell on his way to the stool but finally made his way up to have the Sorting Hat placed on his head.

"Hmm, " said the Sorting Hat in Neville's head, "Interesting. Very interesting. Plenty of potential I see. Perhaps you could reach your fullest potential in Gryffindor."

"Hufflepuff, " thought Neville desperately, "Please put me in Hufflepuff."

"You do have good Hufflepuff traits, " whispered the Sorting Hat in Neville's head, "You are true and loyal enough for Hufflepuff. But you could reach your greatest potential in Gryffindor."

"Hufflepuff would be a great avenue for me to become part of a loyal circle of friends, " thought Neville.

"Hufflepuff is the easy way out for you, " said the Sorting Hat in Neville's head, its whispers becoming fiercer every nanosecond.

"I'm not brave enough for Gryffindor, " though a now queasy Neville, who was fully aware of Gryffindor's reputation as the home of the brave.

"You have the potential for a great kind of bravery in you, " insisted the Sorting Hat, "You just don't realize it yet. Gryffindor will help you to develop and ultimately realize the fullness of the potential you have within you."

"Gryffindors are subject to lofty expectations, " thought Neville, his heart beating a little faster now, "In Hufflepuff I'd be able to be myself without the burden of expectations."

"Your loyalty will show itself in time, " promised the Sorting Hat, "But you want to be in Hufflepuff for the wrong reasons. You want the easy way out. You want the easiest path for yourself. Hufflepuff is a great house that has produced many great witches and wizards, but you would stand out more in Gryffindor."

"How?" wondered Neville.

"You have great bravery within you. But you don't know it yet and it hasn't come close to fully developing yet. Being a Gryffindor would not only allow you to flourish and bloom into being the best Neville Longbottom you can be, but it would also allow you to grow into one of the great wizards of your era."

Neville sat frozen still.

"Gryffindor would provide you a path to blooming into one of the bravest wizards of this generation and by so doing, it may actually lead you into heroic moments. You could one day engage in brave and heroic moments that would make you one of the most celebrated wizards of our time. I can sense within you that you could one day be worthy of the sword of Gryffindor."

Neville very briefly flashed a smile that vanished almost as soon as it had come.

"That sounds great, " thought Neville, "But it's not for me. I don't need to stand out. I'm fine being one of many witches and wizards."

"Finding and discovering your true self is far superior to being one of many witches and wizards, " whispered the Sorting Hat.

"Maybe for some, " mused Neville, "But I'm not sure if that's true for me."

"My job, " insisted the Sorting Hat, "Is not only to evaluate who you are right now, but who you can become and how you can get there. Therefore, it is my duty to make you a GRYFFINDOR!"