Toss the coin twice.
Let TH, HT and TT correspond to the three choices.
And if you get HH, just repeat (so it takes 8/3 tosses on average).
If the coin was biased, TH and HT would occur with equal probability.
So you could assign THHT, HTTH and THTH to the three choices, with other 4-toss outcomes rejected.
Or you could assign HTT, THT and TTH to the three choices, with other 3-toss outcomes rejected.