My vote is 10 years as an arbitrary number, or 2 or more console generations as the graphics technology will have (hopefully) have advanced enough to make an impactful difference in graphics quality.

Control schemes have largely not changed since the PS3/360 era, so there would be no point in remaking a game solely for that, at least not if it's from around that era. For anyone that has played the Rare Replay, Rare Studio's entire collection of games, Jetforce Gemini (a N64 game) had an option in the control settings to make the controls modernized.

All in all, a game should be remade/remastered if it is going to be significantly different to its original form. A fresh coat of paint does not a remake, uh make. For the best example of proper remakes in my opinion, see the Demon Souls remake or the Halo 2 Anniversary edition.