There is a woman that sits daily in the GF lobby. She comes there with a suitcase covered in stuffed animals. We have seen her so many times, we thought she lived there like Eloise at the plaza. Finally curiosity got the better of us and we asked her. She is not a guest, she used to come there with her now deceased husband, and she visits daily to remember him and enjoy the atmosphere. The Gf staff is aware of her, and as of June 2015 she was not asked to leave.
I too have seen her, but never stopped to talk to her. She seems like a pleasant person. However, I have never seen her stretched out on a sofa taking a nap.
I believe the OP said their child was 8 years old, and I'm sure they said they don't use a stroller. I personally think it would be much more convenient to nap and recharge at Hall of Presidents MK, , Ellens Energy in Epcot, and Great Movie Ride in HS. I personally can't think of a quiet spot to re-energize at AK. The Conservation Station can be air conditioned yet it could be noisy. It is quite a trek to a resort to find a restful lobby from AK anyway.