In the US, is it culturally acceptable for a graduate student to give a small gift to a professor?
As your Ph.D. advisor is also going to be your host and provide you with lodging during your stay, a small gift would be very acceptable.  It is accepted etiquette in the U.S. to leave your host a small gift as a token of your appreciation for their hospitality to you.  You may want to give your host a gift card to a restaurant in their area placed inside a thank you card.  This can also be done online and sent to your host via email.  Both ways would be done after your stay is complete.  Another idea would be to send flowers after your visit.  Small tokens such as these that would be enjoyed by the whole household are wonderful ways to say "thank you".