I am not well experienced (yet) but I keep up with the forum so I will start the list with the most common and hope others will chime in. Aquarium size recommendations for all (except where noted) is a 50 gallon (190 liter) or larger tank. Small egged species are almost impossible to tank raise the young.
Atlantic/Caribbean (warm water):
O. vulgaris (semi-nocturnal, can be accilmated to early evening activity, very personable, needs larger tank, recommendations range from 150 t0 300 gallons (570 -1137 liter) , small egged, found almost everywhere in the world. The Atlantic variety seems to be smaller than its Mediterranean cousin).
O. briarius (semi-nocturnal, can be somewhat activated to human hours, very personable, large egged)
O. hummelincki - O.filosus - Carribean Two Spot - recent success with keeping these has been exceptionally poor. They are said to resemble a smaller version of the vulgaris in nature and looks (but have eye spots). Can't help on egg size but if it is a large egg species, this would be the one that would get a lot of attention if it can be tank bred and raised.
O. mercatoris* (fully nocturnal, shy and rarely interact with humans - mine are somewhat of an exception - 20 gallon (75 liter) tank recommended, large egged)
O. joubini (fully nocturnal, look and act like O. Mercatoris but are small egged and not often seen in the trade but the similarities are so stong that it is hard to be sure).
Indonesia (warm water)
A. aculeatus - seems to be the most common arriving from wholesalers this year, the A. (vs O.) is for subgenus Abdopus and typically includes the ability to remove an arm when under attack as well as denoting a smaller intertidal octopus. There are others from the area that show up from time to time but this is the most identified. Muctopus has published several papers/videos on the habit of this octopus to walk bipedally.
Pacific (cold water):
O. bimaculoides* - California Two Spot or Bimac (diurnal, very personable, large egged, identifying blue eye spot - ocellus - considered one of the best to keep as a pet).
O. bimaculatus - (similar to the California Two Spot and hard to distinguish between the two:
http://www.scamit.org/taxonomic_tools/obimac.pdf )
O. rubescens - Ruby or Red Octopus (nice aquarium size, 50 gallon min tank,
requires chiller, can't help with egg size, rumored to be hard to find in the trade because a certain Berkeley Professor acquires most that are available ;>)
* Successfully bred by TONMO member within the last two years.
Not Recommended - Octos with unknown wild population numbers or have other reasons not to keep them as pets:
Blue Ring - deadly toxic
Mimic - unknown population numbers
Wonderpus - unknown populatoin numbers
I hope that gets your research started - we would love to have an octo breeder and would cross fingers for successful importation