In the uncountable, the least cardinal $k$ such that $X$ embeds into $I^k$ is simply the weight of the space $X$, i.e., the least size of a basis of the topology. A collection $\mathcal B$ is a basis for a topology $\tau$ if $\mathbb B\subseteq\tau$ and every et in $\tau$ is the union of members of $\mathcal B$.
In the light of mathahada's comment, it is interesting to note that this notion of dimension actually makes sense for zero-dimensional spaces. In the case of 0-dim spaces you might want to consider embedding in to $2^k$ rather than $I^k$, though. Now, zero-dimensional spaces are called zero-dimensional for a reason, but I don't think that giving $2^{k}$ dimension $k$ is necessarily pathological.
In any case, for spaces that don't embed into $I^{\aleph_0}$, i.e., for spaces of uncountable weight, you have defined something like a local weight that I haven't seen before. (Not that this would mean anything.)