I'm only doing this because none of the answers reflect exactly what I feel is the proper rationale behind the best course of action for anyone at this juncture.
I ended up choosing the Black Star over Aranea, and I regret it. Not excessively, but if I could go back and redo the decision, I certainly would. My rationale:
Soul gems are definitely essential to playing effectively at higher levels. By and large, unenchanted weapons don't cut it unless you're exclusively a melee-type, and even then, they're almost always less effective than their enchanted counterparts. Obviously, then, one needs a supply of soul gems to replenish their depleted enchanted weapons. But I've found that after obtaining the "Soul Squeezer" perk, which gives you much more "recharge" per soul gem, higher level souls are almost always overkill. I can recharge the majority of the enchanted weapons I use regularly with only a Common Soul Gem; Greaters are only rarely required and Grands almost never. Thus, having a refillable Grand Soul Gem isn't really necessary at higher levels from a replenishing standpoint, because Common Soul Gems are easily found in loot and whatnot, and filling them (or even buying filled ones) is pretty easy. Actually, Soul Squeezer boosts up the recharge potential of Petty Soul Gems enough that you can just use about 5 or 6 of them to replace a Common, and finding and filling Petty Soul Gems is ridiculously easy.
Access to money also factors into the equation. After reaching a middling level, having sufficient money is rarely an issue. Buying filled soul gems is therefore a pretty viable strategy.
That being said, I can see how handy the Black Star would be if you're going to be doing a lot of enchanting. I use mostly enchanted weapons I find; unless you have insanely high Enchanting (I don't), what you find in the wild will almost always be better than what you can make. At the end of the day, though, when I want to create something, I'm going to want to use a Grand Soul to make it–it's going to be stronger that way. So if you're making things all the time, the Black Star is the way to go.
I don't dispute the fact that the Black Star, in terms of "fillability," and therefore use, is lightyears ahead of Azura's Star. Humans are far easier to come by and kill than mammoths. It's just that as a sort of battlemage (a mage that uses enchanted melee weapons, but not exclusively), I would far rather have the strongest magic support character in the game than a nonexpendable Black Soul Gem. All of the melee followers I've had get in the way of my area-of-effect spells, and are generally a giant hassle to go questing with.
So, for those considering these alternatives: I'd say that unless you want to be doing a lot of enchanting, go with Aranea. It's true that the Black Star is the only reusable Black Soul Gem in the game, but don't lend that fact excessive weight. There are ways to get around not having it, ways that come as a natural consequence of playing the game. There are decidedly fewer ways to compensate for Aranea: she fills a pretty unique niche.
Lastly, if you're at a low level, don't make this decision: until you've reached at least a middling level and clearly differentiated your playing style, pigeonholing yourself too early would be a bad idea. Mainly because, as someone pointed out, followers don't level. Plus, at low levels, having Grand Soul Gems isn't very useful--you don't have the Enchanting required to make use of them.
