What is $\dim_\mathbb{F} (R/I)$ if $R/I$ is a graded ring and $I$ is homogeneous? If $I$ is a homogeneous ideal, when is $R/I$ a graded ring? And if $R/I$ is a graded ring and we think of it as a $\mathbb{F}$-vector space, is the $\dim_{\mathbb{F}} (R/I)$ always finite and if it is can we find it?
I think that is (and could definitely be wrong!) finite.
For example, if $I \subseteq R=k[x_1,\dotsc, x_n] $ and $I= \langle x^{\alpha} \mid \alpha \in \mathbb{N}^n$ and $\sum_{i=0}^{n} \alpha _i = m\rangle$ ($I$ is homogeneous), then the $\dim_k (R/I) = \sum_{i=1}^{m-1} {n +i-1 \choose i-1}$.
However, I am not sure how to tackle this problem for a general homogeneous ideal, $I$.
 A: If $R$ is a graded ring with homogeneous ideal $I\subset R$, $R/I$ is always graded. Recall that a graded ring $R$ decomposes as abelian groups as $R=R_0\oplus R_1\oplus R_2\oplus\cdots$ where $0,1\in R_0$ and $R_iR_j\subset R_{i+j}$. An ideal $I$ is homogeneous iff it admits a decomposition as a graded module, $I=I_0\oplus I_1\oplus I_2\oplus\cdots$ where $R_iI_j\subset I_{i+j}$. Taking the quotient $R/I$ gives you $R/I\cong (R_0\oplus R_1\oplus \cdots)/(I_0\oplus I_1\oplus \cdots)\cong (R_0/I_0)\oplus(R_1/I_1)\oplus\cdots$, and $(R_i+I_i)(R_j+I_j)=R_iR_j+R_iI_j+I_iR_j+I_iI_j\subset R_{i+j}+I_{i+j}$ which satisfies the definition of a graded ring.
Side comment: in order to talk about dimension as an $F$-vector space, you need $R$ to be an $F$-algebra. Not every graded ring is a vector space over a field- think of $\mathbb{Z}[x,y]$ with the standard grading.
If your ring $R$ is in fact an algebra over a field $F$, the dimension is not necessarily finite. Consider $R=F[x,y]$ and $I=(y)$. $R/I\cong F[x]$ which is certainly not finite dimensional. If $R/I$ is finite dimensional over $F$, we can use the graded ring decomposition to see that the dimension has to be $\sum_{i=0}^\infty (R/I)_i$. So right away we see that $(R/I)_i$ has to be finite dimensional for each $i$ and that there must be some $i_0$ such that for all $i>i_0$, $(R/I)_i=0$. Clearly, if $R$ is already finite dimensional over $F$, these conditions are satisfied trivially, but if $R$ is not finite dimensional over $F$, you have a few things to check.
I don't know general techniques for computing the dimension of $R/I$ as a vector space, but I imagine that computer algebra systems such as Magma or Sage could be used to do the more difficult computations that resist by-hand efforts.
