If you had a relationship with the dog - a particular memory, for example, then a note would be fabulous.
Randomly, the things that touched me most when our dog passed away was the lovely condolence note written by his dogwalker (among other things, calling him the "Paul Newman" of the neighborhood because he was such a handsome senior), and a Facebook post from DH's friend along the lines of "X was a great dog. I believe I'll be seeing him alongside a mountain trail again someday." This was perfect because that is exactly who our dog was - he loved being in the woods and hiking, and DH's friend knew that.
The other stuff is really nice, too. But those things really stuck with me.