The hint to look at A choosing HHH didn't seem to help, so I'll walk through that case.
If flips 1, 2, and 3 are all heads, A wins. Nothing B chooses can come up sooner. P=1/8.
Assume the first time three heads in a row comes up is on tosses n, n+1, and n+2, with n>1.
What was the result of flip n-1?
Won't flips n-1, n, and n+1 be a threesome that will always come up before HHH in the 7/8 of the time that HHH is not the first three flips?