In KR, you should have a pet bar that can be toggled out or shrunk into a button. By default it usually shows up along the left side of the screen. Click the button to fly out the pet bars. (If you're riding your pet, you might have to dismount to see the button initially). When the pet bars fly out, it'll show your pet's name. Click inside the name and a cursor will appear, making the name editable.
You can't rename from the bars that you click-and-drag off of your pet - only from the built-in pet bars.