Is a random circuit likely to compute a one-way function? I remember reading somewhere that (under certain reasonable assumptions) a Boolean circuit with many inputs and outputs (assume equal number for now) chosen at random will be a one-way function with high probability.  Is this true?
The claim that you remember is to good to be true. It is yet unknown (because the question P ?= NP is still open) whether there are one-way functions. If the claim you remember was true, it would prove the existence of one-way functions.

P.S. While P = NP excludes the existence of one-way functions, even with P != NP the existence of one-way functions is still an open problem.