Great question! This is not a complete answer, but hopefully it gets the ball rolling . . .

> **Theorem:** $\mathfrak s_{-}\geq \mathrm{add}(\mathcal M)$, where $\mathrm{add}(\mathcal M)$ is the smallest size of a collection of meager sets whose union is non-meager.

*Proof:* Let $X$ be a set of reals with $|X| < \mathrm{add}(\mathcal M)$, and let $\varepsilon_n$, $n < \omega$, be a sequence of numbers. We want to show how to cover $X$ with intervals $I_n$ of length $\varepsilon_n$.

The basic idea is to use the following fact about Cohen forcing: *in a Cohen extension, the set of ground model reals has strong measure zero*. This was proved by Martin Goldstern in [his first-ever MathOverflow post][1]. Here are the details for the rest of the argument.

Let $M$ be a countable transitive model of set theory with $X \in M$, $X \subseteq M$, and the sequence of the $\varepsilon_n$ in $M$. (Here a "model of set theory" means the usual thing -- only finitely many of the ZFC axioms, but enough to make my argument work!) Such a model $M$ exists by the Löwenheim-Skolem-Tarski Theorem.

Because $|M| < \mathrm{add}(\mathcal M)$, there is a real $c$ that is Cohen over $M$. This uses the fact that *a real is Cohen over $M$ if and only if it is not in any meager Borel set coded inside of $M$*. Because fewer than $\mathrm{add}(\mathcal M)$ meager Borel sets are coded in $M$, there is (by the definition of $\mathrm{add}(\mathcal M)$) some real missing them all.

By Martin's argument that I linked to earlier, in $M[c]$ we have "the set $X$ is strong measure zero". But I specifically put the sequence $\varepsilon_n$ into $M$, which means that this sequence is also in $M[c]$. Thus, in $M[c]$, we have "there is a sequence $I_n$ of intervals of lengths $\varepsilon_n$ covering $X$". But this statement is upward absolute: if it's true in $M[c]$, then it's true. **QED** 

I'll add two more observations relating $\mathfrak{s}_{-}$ to cardinals in Cichon's diagram:

> $\mathfrak{s}_{-} \leq \mathrm{non}(\mathcal L)$, where $\mathrm{non}(\mathcal L)$ is the smallest size of a non-measurable set.

*Proof:* Every non-measurable set is not strong measure $0$. **QED**

> It is consistent that $\mathfrak{s}_{-} < \mathfrak{b}$.

*Proof:* As you mentioned, Laver proved the consistency of $\mathfrak{s}_{-} = \aleph_1$. This holds in what has come to be known as "the Laver model", where it is also known that $\mathfrak{b} = \mathfrak{c} = \aleph_2$. **QED**


  [1]: https://mathoverflow.net/questions/63497/cohen-reals-and-strong-measure-zero-sets