https://www.gearbest.com/tv-box-mini-pc/pp_562952.html (2G ram)
(youtube reviews seem to imply ram is usually upgradeable, with laptop DDR3 sticks you may have lying around)
or just go straight to 4G ram
https://www.gearbest.com/tv-box-mini-pc/pp_635382.html (4G ram)
These are intel boxes, support linux better than most laptops, have slow SSDs (which are still far faster than normal disks, and also usually upgradeable).
They are especially far, far faster than most boards listed here, and they support a lot more software. (including portal 2 which is incredible and on sale: http://store.steampowered.com/app/620/Portal_2/ )
I have no clue, though, if their GPUs support 4k. But they are small, silent, and cover a relatively wide range of performance. Low-end models start at € ~120 (NOT including HDD/SSD!)