My brother, sister, and I were all born with major heart defects, mine being the most severe. While I have a giant team of doctors who work with my case (some of whom I share with my brother and sister), we have one cardiologist whom we've been seeing since we were very young who we've given gifts to a few times.
We've sent thank-you notes a couple times and pictures somewhat often (this cardiologist's office actually has a patient picture bulletin board for the patients who did send pictures). He got copies of my brother and my graduation pictures, and will probably get one from my sister too.
We've also sent his favorite chocolate chip cookies, though we don't do that as much now. When we were all much younger, he was very involved in our lives with surgeries and procedures; now, my primary condition is no longer treatable and my brother's and sister's primary conditions are mostly corrected (all different conditions), and we don't see him as much. During the several year period in which between the three of us someone was always prepraring for or undergoing some type of surgery or procedure, he received more cards/notes/pictures/cookies from us. We used to joke that when we needed him to do something for us, we'd send him cookies and suddenly it would get done.
I also remember being told that, during the time after my last surgery, my grandmother would bring all sorts of treats to the PICU. I was out for ten days after the surgery (problems getting me off the ventiltor), and my grandmother apparently brought something for the PICU staff every day. I do remember in the days I was in the hospital after I woke up that she brought treats often, too.
I don't think we've ever actually sent a gift or a gift card. But pictures, notes, and homemade/edible treats have always been welcomed by my doctor.