Are there any particulars in the specs that would help me determine if a particular yo-yo will be more or less stable than another yo-yo? For example, would a heavier yo-yo be more likely to be more stable than a lighter one or would a wider yo-yo be more stable than a narrower one? Any other things to look at or is there really no way to tell other then by reading the description of the yo-yo? 
Cheaters guide to science:
Look for beefy rims. More weight out wide, less in the middle.
There are some yoyos that are very wide, narrow, light, and/or heavy, all of which may be stable. Ask questions on here if you want to know what people think about them, look at the description, and, as Ben said, look for the beefy rims.
Especially if there is a stainless steel ring around those beefy rims 
Asking around is probably the best way to do it though.
Just be aware, people have slightly different ideas of how much stability is too much.
I’m debating between the Prescription, Terrarian, Manatee, and Set Sail
I owned a prescription and I personally would not call it very stable. The others I’m not too sure on (though I did own a manatee).
Yeah, I don’t know much about the ones you mentioned, but the Too HOT, horizon, TopDeck, and QV are good throws that are on the stable side.
I think anything with 7075 should be more stable theoretically.
Width and rim weight are factors. But it’s a double edged sword. A wide yoyo requires a good throw or it will wobble due to poor technique. The YYJ x-convict is a good example. On a good throw it is rock solid. A little off and it will wobble a bit.
If you want stable get a Masamaxx.
Heavy, weight focused on the rims, narrow, large diameter - are all factors that help a yoyo become more stable. I don’t own a massamax, but from what I can tell it hits all of these factors out of the park and I can only assume it is very stable.
Stable generally means the yoyo is less likely to go off axis. For a yoyo to achieve this type of stability, the most used methods are a larger diameter, medium width, lower walls, and massive amounts of rim weight. A yoyo does not need to be heavier to be stable, look at the Draupnir. A string centering bearing also helps add to stability, despite what many might say. By keeping the string in the center, this eliminates the possibility that the string might cause the yoyo to go off axis by rubbing against the walls. Most commonly used dimensions are 55-56mm diameter, and 41-43mm width.
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Really? I’ve found my prescription to be a stable yoyo.
I think a lot of the difficulty with talking about a yoyos performance stems from the fact that there is not a lot of quantifiable data once the yoyo is in motion.
It’s not like you could say this yoyo is a 5.21 on the Gentry stability scale.
I haven’t played a prescription in a while, but maybe he means stable in comparison to some of his other yoyos.
I consider my Top Deck to be a stable yoyo, but it’s less stable than my Horizon.
So, without a specific comparison, it becomes difficult to label something as stable or not stable.
I compare everything stability wise to my horizon 
Seems like a good place to gauge it from to me
That’s cool, but you are starting near the top of the heap ;D
^I totally spelled gauged wrong in my last post 
Yep, I start at the “top of the heap”, and I’m okay with that. that’s what I consider stable so that’s what I compare to. That’s why when people ask what are good stable yoyos, I can only recommend a few that I believe fit the bill very well yeah there are many stable yoyos that are less stable than the horizon but I honestly have a hard time telling where that would switch to unstable as I go to less and less stable yoyos. Maybe it’s because I’m not observant or maybe it’s because I’m not good at yoyoing, either way I’m going to stick with the horizon as my standard and compare from there.
H shapes that are beefy are a good guideline.
Avalanche, benchmark H (duh)
Weight and design play a part too, design more so.
Example my Envy 64 which is a total Hshape isn’t stable at all, the original Envy isn’t either for me even at 71g, bad design or just outdated maybe.
However I hear the NVX is hella stable
Oh yeah noctus are crazy stable but a little awkward to maneuver, and this is pretty much the 2010 full sized version I’m talking about, it’s a beast…
High rim density, low wall, bigger diameter is more stable, heavier is usually also more stable but at the expense of agility.
That being said, the one affecting stability the most is actually on the other end of the string.
This is absolutely right. Counterweights are what really drive stability.
I think he meant the player. Or maybe you know that and are just being sarcastic.