Why is Harry one of Slughorn's favourite students?

Answer 1

because slughorn really liked harry's mother, lily, as he had once taught her. Lily, as Slughorn says, was really bright in potions and harry is too when he finds the half-blood prince's book.

Also, in general, Horace Slughorn has the habit to surround himself with students who, in his opinion, are likely to become celebrities. This usually gives him benefits later on.