We work in $\mathsf{ZFC}$:
Given an ideal $\mathcal{I}$ of subsets of $\omega^\omega$, let $\kappa_\mathcal{I}$ be the smallest cardinality of a collection $\mathfrak{C}$ of $\mathcal{I}$-positive (= not in $\mathcal{I}$) sets such that every $\mathcal{I}$-positive set contains an element of $\mathfrak{C}$ as a subset. Note that we don't restrict attention to "tame" sets of reals here. The ideals I'm interested in are: the null and meager ideals $\mathcal{N}$ and $\mathcal{M}$ (for the former, note that we can identify $\omega^\omega$ with the irrationals), the dominatable ideal $\mathcal{D}$ of sets of functions which can be dominated by a single function, and the escapable ideal $\mathcal{E}$ of sets of functions which can be escaped by a single function.
It's not hard to show that for each $\mathcal{I}\in\{\mathcal{M,N,D,E}\}$ we have $\aleph_1<\kappa_\mathcal{I}\le 2^{2^{\aleph_0}}$. The upper bound is trivial and the lower bound following from a quick transfinite recursion argument:
Since every countable set is in $\mathcal{I}$, given $\mathcal{I}$-positive sets $(A_i)_{i<\omega_1}$ we can recursively build disjoint $B,C$ which each meet each $A_i$.
Since $\omega^\omega\not\in\mathcal{I}$, at least one of $\omega^\omega\setminus B$ and $\omega^\omega\setminus C$ must be $\mathcal{I}$-positive, and by the previous bulletpoint neither $\omega^\omega\setminus B$ nor $\omega^\omega\setminus C$ contains any of the $A_i$s.
More generally, if $\mathcal{J}$ is any ideal such that $\omega^\omega\not\in\mathcal{J}$ and every set of size $<\mu$ is in $\mathcal{J}$ we have $\kappa_\mathcal{J}>\mu$.
Beyond that, however, I don't see anything interesting. My general question is, "What can we say about the $\kappa_\mathcal{I}$s of these ideals?" This is a bit open ended, so let me focus on the point my ignorance about which is most embarrassing:
For which such $\mathcal{I}\in\{\mathcal{N,M,D,E}\}$ is it consistent with $\mathsf{ZFC}$ that $\kappa_\mathcal{I}\le 2^{\aleph_0}$?
I suspect the answer is "none," and I strongly suspect the answer is "none" if we replace $\le$ by $<$. However, I can't prove any of this. A big obstacle is that it's consistent that there can be an $\mathcal{I}$-positive set of cardinality much less than the continuum; this breaks all the relevant recursions I can think of.
Re: the title, while they have a similar flavor at first sip the $\kappa_\mathcal{I}$s are not genuine cardinal characteristics: cardinal characteristics necessarily live between $\aleph_1$ and $2^{\aleph_0}$, and the $\kappa_\mathcal{I}$s don't. That said, it is very plausible to me that there is a connection between the two topics. Note that the observation beginning with "more generally" above connects $\kappa_\mathcal{I}$ with a genuine cardinal characteristic associated to $\mathcal{I}$, namely the smallest cardinality of an $\mathcal{I}$-positive set.