I've just done a search for game streaming from windows to linux, and it looks like your options are SteamOS streaming (and moonlight if you had a nvidia GPU). I don't know what the licensing on Nvidia's SteamOS stuff and if there's any option of digging this streaming bit out of it and applying it to your favourite linuxes, and how portable it would be come to that.
Also, streaming might be a no go to begin with, because I don't know if it's possible to boot Home Windows without a keyboard and mouse, and carrying around a PC stick, USB keyboard and mouse, Pyra and probably some kind of battery to get this whole thing to run for any time is likely to be impractical.
But if all you really care about are some older games that have had their engines reverse engineered, then a Pyra alone should do you proud.