The Hammer
Its formal name is Mjolnir…and those comic book guys didn't even have to make that odd-sounding name up.
The hammer of Thor was always a part of the figure, going back thousands of years to when bashing people upside the heads with hammers was a lot more socially acceptable. In the myths, it was so heavy that Thor had to use special gloves to hold it. Stan Lee and Jack Kirby altered that slightly to say that only those worthy could lift it.
Here, it becomes a neat little symbol for all of Thor's power. With it, he can literally call down lightning from the sky, as well as bashing giant space robots real good…and even using it to fly. Without it, he's just another underwear model running around broke on the streets of Anytown, USA.
He's a god, but he needs to be worthy of that godhood. And you can see that in Odin's line about the hammer:
ODIN: So long entrusted with the mighty hammer Mjolnir, forged in the heart of a dying star. Its power has no equal, as a weapon to destroy or as a tool to build. It is a fit companion for a king.
Thor's problem is that he takes that for granted. He doesn't understand the responsibilities of the power he wields. He's like a kid with a can of mace, so nuts about what he can do with it that he never thinks about what he should do with it.
But in a flash, it's gone…and he's going to have to work really hard to get that magic hammer back.
And this hammerlessness spurs on a plot, a Hero's Journey, and the story of a giant jerkwad aspiring to be less jerkwaddy.
As Thor soon learns, he's got to do a lot more than just beat up a few SHIELD agents to get to his precious hammer. He needs to learn; to accept the reality of self-sacrifice and to put his money where he mouth is in the hero department.
THOR: Brother, however I have wronged you, whatever I have done that has led you to do this, I am truly sorry. But these people are innocent, taking their lives will gain you nothing. So take mine, and end this.
That's what gives him the ability to make with the lightning and thunder. Not his lineage, but his attitude.