“[...] But his second avenue is nearly as strong: The flesh of his servant, willingly given; the blood of his foe, forcibly taken; and the bone of his ancestor, unknowingly bequeathed. Voldemort is a perfectionist—” Albus glanced at Severus, who nodded agreement, “—and he would certainly seek the most powerful combination: the flesh of Bellatrix Black, the blood of Harry Potter, and the bone of his father. [...]”
This is persuasive, but… why the heck would Voldemort go the trouble of breaking into Azkaban instead of grabbing Snape or something?
In Chapter 61 Dumbledore says:
Right, this is a stronger interpretation.
Reduce, re-use, recycle.
VM said he broke into Azkaban to find out where his wand was; there’s also the flesh of the servant thing. Using her Dark Mark is a secondary benefit.