Seven hat colours and a guaranteed correct guess

Seven prisoners each receive a hat in one of seven colours (colours may repeat). Each sees the others' hats but not their own, and all write down a guess simultaneously without communicating. They are all released if at least one guess is correct. Can they agree on a strategy that always works?

Yes. Number the colours $0$–$6$ and the prisoners $0$–$6$. Prisoner $i$ guesses the colour that would make the sum of all seven hats $\equiv i \pmod 7$. The true sum has exactly one residue, so exactly one prisoner is right.