The following Acer has a Core i5-5200u CPU, 1080p resolution screen and a nVidia 940m which is good enough to play most games at 1366x768 / 1600x900 with low or medium graphic settings. So playing games on it would kinda be like a bonus. Recording video generally relies on the CPU. It costs $510.
http/www.newegg.com/Product/Product.aspx?Item=N82E16834315065&cm_re=acer_940m-_-34-315-065-_-Product
A less expensive alternative would be the following Asus which has a Core i3-5010u and a 1080p screen for $350. It relies on integrated graphics and it is above the minimum requirements for the Elgato for recording purposes.
http/www.amazon.com/dp/B011KFQASE/ref=psdc_565108_t2_B0088L182C
If he wants to stream video, then he will need a more expensive laptop with a quad core i7 CPU like the i7-6300HQ.