a) It was invented by Grothendieck, see EGA I, Springer edition, especially chapter 0, discussion of representable functors.
b) Quite possibly is a shortcut for $Hom$. Sometimes the letter $y$ is used (for Yoneda). The trouble is when you are considering the representable functor defined over several categories, e.g. a category and a subcategory.
Bonus: If you, instead of considering contravariant functors $\mathrm{Sch}^{o} \to \mathrm{Set}$, use covariant functors $\mathrm{Aff} \to \mathrm{Set}$ the notation used in EGA is $h_X^{o}$. Perhaps the reason is that Yoneda's map is contravariant in this case.