I imagine a lot of the selection was indirect selection for neoteny. I think it would be much, much harder to select for domestication in octopi, as they do not raise their young.

Wikipedia on the fox-breeding experiment.

I'd speculate that if you did an identical breeding experiment with octopuses (as in, the breeding criteria of non-aggressively interaction with human hand) you'd breed for curious, bold, or playful octopuses which tend to approach novel stimuli ... but not friendly in the sense of affectionate.

It's not that they're asocial, I think they sometimes lay eggs cooperatively and obviously seek each other out for mating... but primarily octopuses see others of their species as predators or prey. (I mean, cats do eat each other but only in bounded contexts, like infanticide, not hunting.)

octopi

Octopuses / octopodes. It's greek, not latin.

Or could we breed them for intelligence...? With such short periods between generations, we could reach superintelligence, maybe faster than other methods!

I would imagine that using foxes give you a lot more to work with though. Foxes in nature live in pairs or small groups. The children stay around the parent for a long time. So they already have mechanisms in place for social behaviours. (And even if they are not expressed, there probably are some latent possibilities shared among mammals? E.g. this article about the evolution of housecats notes that they independently evolved a lot of the same behaviours that lion prides use to socialise, even though wildcats are solitary.)