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Raspberry Pi runs Kodi fine, there are a couple of different distributions which make it very easy. There are options for audio over HDMI or jack plug or whatever.
Other alternative is an Android box if you think you'll use the extra features. From my experience the cheap ones (in the Raspberry Pi price range) don't run Kodi as smoothly as a Raspberry Pi does so only worth it if you're planning to spend a bit more and use more of the features.
Yeah it'll only be used for Kodi, possibly a bit of internet surfing.... what would you recommend, a Kodi box, Raspberry Pi perhaps? Just struck me if going down the HDMI videocard route then it'll have to support audio over the HDMI port, although I expect they all do that these days?