Centralizers of elements in free profinite groups I believe, although I can't say that I've given a rigorous proof, that for a free group $F_r$, and an element of it $a$, $C_{F_r}(\langle a \rangle)=$ the group generated by the elements $b \in F_r$ such that $a=b^n$ for some integer $n$ (I will say: the group of powers and roots of $a$).
One may similarly ask (and this is my interest in this), given $ a\in \hat{F_r}$ (the free profinite group on $r$ generators), what is $C_{\hat{F} _r}(\langle a \rangle)$? In particular, is it the profinite completion of the group of powers and roots of $a$?
 A: I do not know the complete answer on your question. However it is clear that the situation it the profinite case is different.
Consider a semidirect product   $G=\mathbf Z_3\rtimes \mathbf Z_2$ where a generator $a$ of $\mathbf Z_2$ acts by sending elements of $\mathbf Z_3$ to their inverses. Then $G$ is a proyective profinite group (because all its Sylow subgroups are free pro-$p$) and so it is a subgroup of a free profinite group. Now the centralizer of $a^2$ contains $G$ and so it is not abelian.
Added: If $a$ is a non-trivial pro-2 element of $\hat {\mathbf Z}$ (profinite completion of $\mathbf Z$), then all the roots and powers of $a$ are also pro-2 elements.
A: An example: The direct product of the groups $A$ and $B$, where $A$ is a 2-generated pro-2 group and $B$ is a 2-generated pro-3 group is projective. There is a canonical epimorphism from the 4-generated free profinite group $F$ onto $A \times B$. By the projectiveness of $A \times B$ this group retracts into $F$. Then $A$ contains nontrivial elements with non-pro cyclic centralizer in $F$. 
Wolfgang Herfort, Vienna.
A: Another partial answer:
It is a theorem of Herfort and Ribes that if $A$ and $B$ are profinite groups, and $G = A \sqcup B$ is their free profinite product, then centralizers of nontrivial elements of $A$ lie in $A$. (See Wolfgang Herfort and Luis Ribes, Torsion elements and centralizers in free products of profinite groups. J. Reine Angew. Math. 358 (1985), 155–161. )
So, at least centralizers are nice when $\langle a \rangle$ lies in a cyclic free factor.
