Can we narrow this down a bit, what are you looking for? I do not know gaming, give an idea of processor, speed, memory. Is Nana running a Mac? can you upgrade the memory? Is computer control the only way they can interact with you? If you had your own would you go in your room and never come out?
The best way to get internet is steal it. Any public WiFi around? A Cantenna can reach a mile for a good line of sight signal. Do you have a neighbor you could borrow some bandwidth from? You will show up on the network, so it is best to ask.
Here dialup cost me $25, but a second account cost $12, now some is $10, what is Nana running.
In a lot of places free dialup through the Libuary, and now here WiFi. It depends if you live in a city, or outback on a sheep ranch.
Ebay is soso, I do better local with no shipping. I sell worldwide and shipping to Australia costs, or is slow, two months by boat.
Pentiums, and IIs laptops are cheap, anything from Windows 98 up works, but XP heals itself.
Owning a computer is a minium human right.