Let $R$ be a commutative unital $G$-graded ring , where $G$ is a monoid ; then does there exist a $G$-grading on $R[X]$ such that whenever we have a commutative unital $G$-graded ring $S$ , $a \in S$ and a graded homomorphism $\phi : R \to S$ , then there exists a graded homomorphism $\bar \phi : R[X]\to S$ such that $\bar \phi |\_{R}=\phi$ and $\bar \phi (X)=a$ ? If this cannot always be done , then what if we took $a\in S$ to be a homogenous element ? If even this cannot always be done ; then is there any condition on the monoid $G$ such that this type of grading can be done for any $G$-graded ring ? 



Providing a grading for the polynomial ring over a commutative unital graded ring
This answer is only about the case where $G$ is a (commutative) *group.* I have not thought about more general monoids yet.


As observed by Remy in his comment, one has to choose $X$ to be *homogeneous of some fixed degree* and $a$ to be *homogeneous* as well. Then, this is indeed possible for polynomial algebras with an arbitrary set of indeterminates, or even more generally for algebras of monoids. For details you may have a look at paragraph 2.1.6 in [this article](http://dx.doi.org/10.1016/j.exmath.2013.06.003) (or [here](http://arxiv.org/pdf/1306.1983) for free).


