A traditional, important graph invariant similar to what you are asking about is the genus of a graph.
I am aware that one may argue that this does not capture the intuition of 'dimension'. But then again, you yourself tagged this a 'soft-question', so this answer should be sort-of-acceptable to you.
Also worth pointing out: as you will probably known, for any fixed dimension $d\geq 3$, any countable graph can be embedded into $\mathbb{R}^d$, a fact which by the way can be proved directly by induction (you do not need the usual argument via a Vandermonde matrix).