Palo is nominally for adults only, so I guess the answer would be no. I would imagine a mature-looking 16 year-old wouldn't attract attention, so she may be able to eat with you (he says at risk of starting the same old debate again).
Some other options, besides the main dining rooms, are Pinnochio's Pizza, Pluto's Dog House, room service, and (on the 7-night cruises) alternative casual dining at Topsiders. I don't know if there is food other than snacks at Common Grounds.