First: Make sure you opened the door in the Barrow that requires the claw. The barrow has two bosses, not one (Wounded Frostbite Spider + Draugr Overlord). This leads to an item required for http://www.uesp.net/wiki/Skyrim:Bleak_Falls_Barrow_%28quest%29

Then, if that doesn't work, you can also try Camilla who will also accept the claw.

If that doesn't work and you are on the PC, you can use Console Commands. Here are a few that I use if I get stuck on quests (MS13 is the quest name).

PC: setstage MS13 69 -assigns objective to return claw or setstage MS13 100 (or 200) - completes quest

Finally, if that doesn't work, try resetquest MS13 followed by StartQuest MS13. You can use the setstage commands from there.

First, make sure you opened the door in the Barrow that requires the claw. The barrow has two bosses, not one (Wounded Frostbite Spider + Draugr Overlord). This leads to an item required for the Bleak Falls Barrow quest.

Then, if that doesn't work, you can also try Camilla who will also accept the claw.

If that doesn't work and you are on the PC, you can use Console Commands. Here are a few that I use if I get stuck on quests (MS13 is the quest name in this case):

Type setstage MS13 69, which assigns the objective to return the claw, or to just skip right to having the quest complete: setstage MS13 100 or setstage MS13 200.

Finally, if that doesn't work, try resetquest MS13 followed by StartQuest MS13. You can use the setstage commands from there.