If you went through and underlined every time Tally referred to her scarred hands, you'd probably get a blister because it happens so frequently. But you'd also find that her scarred hand comes up all the time when friendship is on the line:
So we're guess you can tell why we're saying that her scarred hand is a symbol (at least for Tally) of friendship and promises to friends.
Which is also why it's interesting that hands are what sets these friends apart. For instance, David's hands prove that he's a different sort of kid: his hands are "as rough as the wood grain of the table in the dining hall, the skin along his thumb as hard and dry as leather cracking with age" (26.50). Of course, because he has rough (and bigger) hands, he doesn't need his old gloves to work, which is why he gives his old gloves to Tally… which then makes Shay very angry. So hands, the symbol of friendship, eventually end up tearing friends apart. Hello, irony.