For $400 your best bet is the following Toshiba Satellite L55D-C5318. It has an AMD A10-8700P APU with a Radeon R6 (Carrizo) graphics core. It is a little less powerful than the Intel HD 530 graphics core which means it is a little slower than a nVidia 920m dedicated GPU.
Bumping up to $500, your friend can buy the Acer Aspire E E5-573G-52G3 with a Core i5-5200u, nVidia 940m and 1080p screen. The 940m is certainly more powerful than the above mentioned 920m, but it is not really powerful enough to play most games at 1080p. Though I suppose using low graphic settings less graphically demanding games could be played at that resolution. The 940m is best for playing games at 1366x768 / 1600x900 with low / mix of low and medium graphic settings for reasonably good performance.
Bumping up to $700 can get you the following Acer Aspire V Nitro VN7-571G-50VG with a Core i5-5200u, nVidia GTX 950m and 1080p screen. It's better than the above Acer Aspire E laptop for games, but you are now at the point where if you spend $100 you can get a significantly more powerful laptop....
Well... looks like the price increased by $10 the last time I checked. For $810 you can get the Dell Inspiron 15 7559 with a quad core i5-6300HQ CPU, nVidia GTX 960m and a 1080p screen. This laptop is significantly more powerful than the Acer Aspire V Nitro I linked above. As far as most gamers are concerned the GTX 960m is least powerful graphics chip that they would actually consider a "gaming GPU".