In this case, as in almost every case, you should follow the standard rule for determining when to use "the" vs. "a" for singular nouns. If you expect the 聞き手 to know what you are referring to, then use "the" (定冠詞). Otherwise, use "a" (不定冠詞). In this case, obviously the gift-giver knows which gift is being spoken about, so "the" is appropriate. ("Your" is also OK.)
Note, though, that you might see "a nice present" used in a context such as the following:
"Thank you for bringing such a nice present." "such a ..." is a set phrase here.