The demon's gift to the girl was life.
The gift may appear to have been given the moment Tenseiga brought her back from death, but he didn't give it to her the time she died in the forest. Bringing her back was all the sword did. Wasn't he just "testing it out" after all? He could have just left her there and she'd go back to before - the lonely little girl who didn't say a word.
The gift was given every moment afterwards...when Sesshoumaru invited Rin to her life. An actual life. A place where she could laugh, be protected and most importantly have the assurance that there is someone there for her. In her life there is even someone to talk to (no matter how little reply she may get).
Invited? But didn't she follow him on her own?
Sesshoumaru is not one of many words. His way of expressing himself could be considered unique...
Sesshoumaru's invitation is that he hasn't killed you.