I’m guess she just wants to surf the Internet, play solitaire and things that that, correct?
If I were you, I’d get a netbook. An MSI Wind or Asus Eeepc 1000HE would be perfect. They are low on cost($250—$400) and perfect for doing exactly what you want.
As for graphics card, a dedicAted card is better than inegrated, but for your mother’s purposes, integrated is fine.
Edit: You said she watches a lot of online videos. Integrated graphics probaby are fine, but if she tries to watch HD video, it might not work as well. Normal YouTube and Hulu videos should be fine.
RAM, you’ll want at least 1GB. Most come with this standard.
I’d try and get one with a mechanical HDD with at least 100gb space. Many of these netbooks come with low capacity SSDs, which are safer, but have very little space.
A dual core processor has more power than a single core. The dual core will make the computer faster, but most netbools don’t have them. You don’t need a dual core. I expect any netbook to have an Intel Atom 280 processor, with the Intel 945G integrated graphics. This will be more than suitable for her needs.
Sorry for the giant response, but I hop this is helpful. Happy buying, and once again I think you should look into a netbook.