Going down the route of FF characters there's a later game which featured a prominent black character. Kiros, who by the way was the most intelligent and wise member of Laguna's squad.

Involving other games series; Resident Evil 5 had Josh and of course Sheva. Gears of war had Dom, Jace and of course, the one the only Coletrain baby! I'm currently just skimming over the games on my shelf here. Another major game series with prominent black/ethnic minority characters would be the Mass Effect series, Anderson is a black guy, Udina was indian, Emily Wong is asian descent. It goes on. Heck if I remember a character description of Shepherd early on it was supposedly the reason for his tanned complexion (on default shep) being that he was meant to symbolize a mixture of different cultures, same with Ash in ME1. In ME2 and 3 it seems Shep and Ash got more white. I would mention Assassins Creed 1 here too since it's set in the Holy Land during the crusades...

I think that you're wrong about the inclusion of ethnic minorities in games, sure some games don't include many ethnic minorities for some reason or another, though notably because of the setting. At the same time, some games provide very few white characters (hello Resident Evil 5) but then again look at the setting, everyone screamed racism about RE5 but it was taking place in Nigeria... if all the zombies had been white that'd be more racist than anything else in the game.