As Bruno rightly points out, my first answer (below) is nonsense. So let me try to say something else vaguely useful.
In the Spring, I heard Ian Biringer talk about his recent work with Juan Souto. I'm pretty sure he stated the following theorem:
Theorem: Let $\epsilon>0$ and $r\in\mathbb{N}$. There are finitely many hyperbolic 3-manifolds with boundary $M_1,\ldots,M_k$ with the property that any hyperbolic 3-manifold $M$ with injectivity radius greater than $\epsilon$ and $\mathrm{rank}\pi_1M\leq r$ can be obtained by gluing the $M_i$ together along their boundaries.
This would imply that there is a finite generating set of blocks for hyperbolic manifolds with appropriate restrictions on the injectivity radius and rank. I believe the proof is non-constructive, so one doesn't actually know what the $M_i$ are.
Poking around on the archive and Ian's web page, I don't see the result in question, so I'm at a loss to provide a reference! But if this sounds useful, then no doubt one could contact Ian or Juan and get the details.
I'm somewhat out of my comfort zone here, but I think this is right.
The figure-8 knot complement $M_8$ is universal, meaning that every closed 3-manifold arises as a Dehn filling on a finite-sheeted covering space of $M_8$. So the family $\lbrace M_8, D^2\times S^1\rbrace$ generates all 3-manifolds. So the family $\lbrace M_8, D^2\times S^1\rbrace$ generates all 3-manifolds.
I don't have time to look at the references right now, but I'll try to get back to it later.