How can I personalize a Radio Flyer spring horse for a child?

I bought my friend a Radio Flyer spring horse for his new baby (knowing that she won't be able to use it until later in her life). I want to get the gift personalized, and there are locations where it could be embroidered, painted, etc..., but I'm concerned that one usually puts a horse's name on a saddle, and therefore it would be awkward to personalize the saddle, or any other part of this gift, with the baby's name, including the frame. I'm looking for suggestions on if I should personalize the gift, and if so, where to put it.