$650 is pretty difficult. I looked into what it would take to replace my 4 year old gaming laptop. The current minimum I could spend to do so is 500 bucks and that would be for a lenovo with a high end APU. However my current laptop can barely play Star Trek Online. Plays Bad Company 2, Crysis 2 as far as FPS goes. General range for a gaming laptop is around $750 to $1500 bucks depending on if you want something that can play 2 year old games, or this years games (and probably not next years...).
I usually use notebookcheck to get a general reference for the cards -
http://www.notebookcheck.net/NVIDIA-GeForce-GTX-780M.88993.0.html The basic scheme for dedicated GPU's on the nvidia side is the mobile model is the same spec as 1 below its number on the desktop side. The 780M is basically an under clocked GTX 770. So on and so forth. For serious gaming I would never go below the 650/750M, 660M+ being optimal but probably out of the price range.
I haven't kept up on the AMD end but typically they will have something like a 6990M or 7990M as the top end and its a single card not a double
Quite honestly in the budget range she is in looking for a strong AMD APU might be the ticket. Its what I have been looking at. Integrated GPU's will give her much better battery life but the AMD numbering scheme is pretty messy on the APU side. Just make sure you look it up and compare it to what performance you think is needed if you go that route.
Not recommending this but to give you an idea of what I have found on the low end of the price scale,
http://www.newegg.com/Product/Product.aspx?Item=N82E16834313719. The 8650G is roughly in the same neighborhood as my old 9800M GS, but then again I flashed mine to a 9800M GTS and then overclocked it into oblivion. Id prefer something with an i7 if your going dedicated graphics, refurb ASUS G75's go for under 900 bucks.
1366x768 is pretty typical for her price range and for windows 8 laptops. If you can get a decent GPU I would shoot for at least 1600x900 that way work isn't so bad. Im holding out for a 1080P screen but obviously can't game at that res on most of these gpu's.
