Real talk: if you killed a bear the size of Shamu and lived to tell the tale, you'd be bragging about it to every person you ever met. And if you ever do ever find yourself in that situation (the odds are not high), then you'd be well-served by following Glass's example and using the flashiest part of your victim as a trophy: its hand.
For obvious reasons, the bear claw represents Glass's strength and prowess. Easy enough, right? Glass adds to this meaning by attaching the claw to the hawk's-feet necklace he received from a Pawnee chief, which gives him a sense of pride, knowing "that the claw that inflicted his wounds now hung, inanimate, around his neck" (1.10.53). The claw does send the right message: Yellow Horse, for example, feels a great deal of respect for this odd white dude when he spots the dude's trinket.
Ultimately, Glass gives the bear claw to Yellow Horse as a way of thanking him for saving his life. It's a powerful gesture, and one that we're sure Yellow Horse appreciates.
Before we close this thing out, we have to mention the fact that Glass wouldn't have had this claw at all if it weren't for Bridger. True story. After the attack, Bridger had slipped the claw into Glass's bag while the other trappers hacked up the beast for themselves; that's an act that shows Bridger's compassion for Glass.
Hugh Glass's bear claw: making your elementary school participation trophies seem insignificant since 1823.