Let $R$ be a henselian ring. Is $R[[x]]$ also a henselian ring?
Yes, it is. More generally, the following result holds.
See N. Sankharan, A Theorem on Henselian Rings, Canad. Math. Bull. 11 275-277 (1968), in particular Corollary 2.
Remark. The Proposition above is no longer valid if one takes the polynomial ring instead of the power series ring. For instance, if $K$ is a field than $K$ is henselian but $K[x]$ is not.