The Oxford English Dictionary
records the phrase "jury-rig
ged" as far back as 1788
, with "jury-mast
" and other uses of "jury
" for a temporary replacement for a damaged part of a ship
as far back as 1616
The phrases "jerry-rigged" and "jerry-built" derived from "jury-built" and it is quite possible, as stated in the node for "jerry-built," that this derives from "injury" or Old French "ajurie" which means "relief," however the OED states straight out that there is no proof of either of these theories.
Mentioned in both "jerry-rigged" and "jerry-built" is the theory that "Jerry" refers to Germans; it is true that that slang nickname existed, but the OED does not record it earlier than 1919, so it is not really possible that the phrases could have come from the belief that Germans did not do good work.