What type of gift could I give to my PhD advisor?
As far as I'm aware, giving gifts to your PhD adviser upon completion in the US is not an obligatory ritual, as compared to Christmas gift exchange in families, or Mother's Day cards and flowers, which are obligatory rituals.

Before you decide to give a gift and choose one, you should consider your adviser's personality, value system, relationship style, and cultural background. Does this adviser frequently give and receive gifts from other faculty members?  From other students?  Does he/she display many gifts or honors in the office?  If so, examine the symbolic nature of these gifts.  Are they personal to him/her?  Are they status oriented?  Do they reflect or signify certain values or identity?  Following this line of inquiry is fruitful on two levels.  First, what ever you give him will certainly show that your knowledge and appreciation of him/her as a unique person.  Everyone likes to be valued for their unique, special qualities and "quirks".  Second, you will be giving something that has significance in the context of your relationship with him/her, not something that is generic.

My personal opinion is that the best gift a PhD student can give their adviser is to be a great student, to be someone who is easy to advise, to be someone who heeds the advice, to be someone who stimulates the adviser intellectually, and, above all, finishes the PhD program on time (or at least expeditiously).  You can also express in words what your adviser means to you in your acknowledgements section of your dissertation.  You can also offer public thanks at the dissertation defense.

If you think a gift is warranted above and beyond this, then I'd lean toward something that has low economic value but very high symbolic value in the  context of the adviser as a unique person and your unique relationship with him/her.