From Tony Thorne, The Dictionary of Contemporary Slang (1990):
bog-standard adj British totally unexceptional, normal and unremarkable. Bog here is used as an otherwise meaningless intensifier.
From Paul Beal, Partridge's Concise Dictionary of Slang and Unconventional English (1989):
bog-standard. Standard, straight from a factory, with no refinement or modification: orig. applied mainly to motorcycles, since mid-1950s; by 1980s > gen. engineering, with wier application. Prob. ex bog-wheel. (Bishop; Hanley, 1988.)
...
bog-wheel. A bicycle: Cambridge undergraduates': earlier C.20. (Keynes, 1981, quoting his diary for 1907.) Its wheels are—like the gap in a water-closet seat—round. Cf. bog n., 1.—2. ["Abbr. bog-house, ... a privy, since early C.19 (in Spy, 1825); orig. Oxford University s{tudents}"; "Abbr. bog-wheel, a bicycle: Marlborough College"] Hence, a motorcycle: Army: WW2. (L.S. Beale.)
So Oxford University students were calling privies "bog-houses" by 1825, and Marlborough College and Cambridge University students were calling bicycles "bog-wheels" by 1907—presumably because of a perceived similarity between the roundness of bicycle wheels and the roundness of the open space in a privy seat. From bicycles, the term bog-wheel was extended to motorcycles, and by World War 2 the term was sometimes shortened (when used in that sense) to the word bog.
The compound adjective bog-standard then arose in the 1950s in the sense of "standard-issue," first in connection with motorcycles, but later applied to other machinery, and eventually (it seems) to such general things as scoops of ice cream.