A gift? Like a present? From whom?
The thing Peter seems to want most in the novel is power. He is a genius, he doesn't really need any material possessions (at least it isn't mentioned that anyone seems in need of material possessions - the kids take nothing to battle school - and seem to have all the latest technology at their fingertips anyway), he's successful in school without studying, and he is fooling nearly all of the adults in his life.
At the end of the book - after Ender is successful in leading the battle at the Bugger home world - every major country back on Earth wants Ender. If Peter's ultimate goal is to be the leader of the world, perhaps a good gift for him at the end of the book would be his brother. (*Of course this would be dangerous in more ways than one.)
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Now if you mean a gift like gifting - a personality strength - then you could answer that Peter's best gift is the art of genius manipulation. He is gifted in figuring out what others fear most - and using it against them (I believe this is in Chapter 9). He comes across as a brilliant child and model student in school - he even appears to have his own parents wrapped around his little finger (a theory that is debunked in the 3rd book of the series Shadow of the Hegemon). Ironically, what he wants most in life is more and more power - and probably what he is most gifted in is actually achieving power, through manipulation.
That depends. Do you want to give Peter what he wants or what he needs; because the two are very different.
Peter is ambitious for power. If you can gift him a Hegemony seat or something along those lines, he's sold. Peter is intelligent, cunning, and ruthless. Material goods will offer him very little that he could have obtained for himself otherwise. Giving him a position of authority over others is a perfect gift.
However, if you want to give Peter what he needs, then you should get him some psychiatric help for his vicious violent ruthlessness. Or maybe a self-help book.
What a fun assignment!