A cute helmet bag, grooming kit, or a backpack for her "barn stuff" would probably be great, and she can use them for multiple horses. A saddle pad would also be great, or her own set of reins for lessons would also be cute! I wouldn't go too far beyond those just simply because she will eventually move on to different horses, and after a little while you run into issues of fit.
Ettiquete-wise, I can't see any trainer who works with small children being upset by a small child wanting to use something "they" picked out, or got for Christmas, on their lesson horse except where it might affect safety for the rider or the horse. Anyone who has worked with small kids knows that you gotta pick your battles :)