It is still a very new hobby animal and we know little about octopuses in general. In recent years, work is being done on growing them out for food and I am hoping the pet world will benefit from some of these studies. The most scientific work has been done with O. vulgaris (mostly outside the US). Unfortunately, they are often larger than most can accommodate as pets (although LittleBit was just about right for a 65-75 gallon tank, had she gotten any bigger, I had a 140 alternate but most people don't) and they are small egged so raising young is a long way from being practical. O. mercatoris is a nice size but very shy and nocturnal. However, they are large egg and likely the easiest to raise. Mucktopus has a series of publications on A. aculeatus, and it is a good size but most are adults that come in from the Philippines and don't live long (also a small egg animal and often has hatchlings in keepers' tanks).
At the present time, if you have the room for O. briareus, it is the most successful IMO. We are starting to see more and more young ones, they are crepuscular as they become juveniles (pretty much nocturnal until about 5 months old) and seem to survive through senescence in an aquarium (65+ gallons needed, CaptFish will argue that this can be too small but most do not get as large as his Legs). They are an abundant Caribbean species and usually come directly from collectors (either caught directly or as by-catch from the crabbing/lobster industry) so they miss the wholesale handling and overseas transport. O. briareus is also a large egg species so there is the possibility of raising a few hatchlings (this is not easy to do for any species so don't take the possibility as will happen but know that it has).
There are other possibilities that we just do not see in the trade. If they become a bit more main stream, we may see others over time. Somewhere along the line I think I decided I wanted to keep at least one of every species I have seen (GPO excepted)
I have kept one that I will never know for sure its ID unless it starts showing up more often (Monty was an excellent aquarium animal) and that was probably my biggest frustration. I have one now that I think is not what I initially guessed and won't know if he is what I now think he is until he gets bigger
. Fortunately, I DO know where he came from so his environment should be satisfactory.