I googled the words "to sock away" and came up with definitions aplenty, but no reference to the phrase's origin. Surprisingly, at least to me, was the suggestion, from Oxford, that it was a North American usage. The Free Dictionary adds New Zealand to that small cluster.
I tried to imagine people hiding their money in a sock, either worn or hidden somewhere else. What else to do with a single sock, once its partner has skipped town? Then I remembered that Norwegian has the term "å sokke bort" which seems remarkably similar.
This information wasn't completely satisfying. Why not just "sock it" or "put in a sock"? And did the Norwegians get it from English or vice versa? Or has it spread from/to elsewhere?
I don't know if it is just me, but the idea of socking something away does not sound like a gentle motion. Perhaps with a little sleight of hand - so everyone doesn't know where you've put it. That makes me wonder if it is related to "socking s.o" as in "I socked him in the eye". It really forces me to think that this has nothing at all to do with socks.