I think it most likely stands for spatium. E.g. Euler’s first book Mechanica (1736) uses $s$ throughout and first introduces it as follows (p. 13):
Theorema. (...) oportet determinare tempus, quo arcus $\mathrm{AM}$ absolvitur.
Solutio. Sit spatium $\mathrm{AM}$, sive sit linea recta sive curva, $=s$, et celeritas, quam corpus habet in $\mathrm M$ sit $c$ (...). Integrando ergo habebitur tempus, quo totus arcus $\mathrm{AM}$ absolvitur $=\int\frac{ds}c$.
(I don’t know who started this. A letter of Leibniz to Huygens (1690) has “resistence $r$, vistesse $v$, temps $t$, espace $s$”. Johann Bernoulli’s work on caustics (1692) has spatium curvilineum (p. 57) but apparently not the letter $s$, which he uses in later papers (ibid. pp. 60, 309, 316, 409).)