I think the scene in Fellowship defined it for me, when they all were sitting in council in Gondolin, around the Ring on a pedestal. All the bickering and infighting between the races flares up, and if you pay close attention, none of them actually make an argument for themselves taking the Ring to Mordor, only arguing against any of the others taking it. Finally Frodo cuts in, at first quietly but then speaking up over the crowd, that he'll continue the journey. He was only asked by Gandalf to go as far as Gondolin, but he volunteered his life to go even further, to do what none of the other races could be trusted to do. And in the end, though he was finally overpowered by the influence of the Ring - and honestly who could blame him, everyone gets corrupted by the ring - he made it all the way to within inches of tossing it into the mountain.
As for the other hobbits - Sam included - they only tagged along because Gandalf had to threaten them. Sure, they went along with the Fellowship after Gondolin, but if they hadn't been coerced in the beginning, none of the after-events would've happened. The only one volunteering themselves from beginning to end, the only true hero was Frodo, I say.