A $\sigma$-algebra generated by $\kappa$ sets (for $\kappa>1$) has at most $\kappa^{\aleph_0}$ elements (see cardinality of the Borel $\sigma$-algebra of a second countable space). It follows that the same is true for functions as long as $\kappa$ is infinite: a $\sigma$-algebra which is initial for by $\kappa$ functions to $\mathbb{R}$ (in the sense you describe) has at most $\kappa^{\aleph_0}$ elements (since the each function contributes countably many generating sets, namely $\{x\in\mathbb{R}:f_i(x)<a\}$ for $a\in\mathbb{Q}$).
The Lebesgue $\sigma$-algebra has $2^{2^{\aleph_0}}$ elements, since for instance it contains all subsets of the Cantor set. It follows that it cannot be initial for any family of $\kappa$ functions where $\kappa^{\aleph_0}<2^{2^{\aleph_0}}$ (in particular, this includes $\kappa\leq 2^{\aleph_0}$).
I see no reason to expect there is any nice "natural" or "minimal" set of functions you could use. (Indeed, with the usual meaning of "minimal", I would expect that there does not exist any minimal such set at all, though that seems rather difficult to prove.)
Note that this question is basically the same as asking for a set of generators for $\mathcal{P}(\mathbb{R})$ as a $\sigma$-algebra, by picking a bijection between $\mathbb{R}$ and each uncountable Borel null set.
