Hi Chloe,
How wonderful that your mom has offered to watch your son while you and your sister explore Walt Disney World! I remember how fantastic it was not only to go to Walt Disney World with my daughter when she was an infant, but also how nice it was when someone watched her for a while so I could spend time with the other adults.
There are some great places for your mother to take your baby at Walk Disney World where she does not need to pay admission. The first that comes to mind is the newly re-imagined
Disney Springs. This shopping, dining and entertainment area is gorgeous. This is a lovely place to walk around, do some window shopping, and even browse in the stores.
Another area with shopping and dining, but that is smaller and more quaint, is
Disney's BoardWalk. This area is smaller than Disney Springs but has a lovely area to stroll around Crescent Lake. Disney's BoardWalk is located within walking distance or a short boat ride of both Epcot and Disney's Hollywood Studios, so it would be a good option if you and your sister were at either of those theme parks. In fact, the boat ride is quite relaxing and would be a nice way for your mom and baby to spend some time.
Finally, do not forget the resorts! If you are staying at a Disney resort hotel, your mother and son can relax there. Another option would be to ride the monorail from resort to resort and explore the shops and lobbies in the resorts along the monorail line. I remember once when my daughter was an infant she fell asleep in her stroller on the monorail, and we just rode the monorail around and around for an hour until she woke up. It was relaxing for her - and us!
I hope that these ideas are helpful for you, Chloe. Please do let us know if you have any further questions!