Out of curiosity checked Merriam-Webster, probably more than you want to know...
" Jury-rig comes from the adjective jury, meaning "improvised for temporary use especially in an emergency," or "makeshift." It's a 15th century term that comes from the Middle English jory, as known (back then, anyway) in the phrase "jory sail," meaning "improvised sail."
The rig in jury-rigged likewise has nothing to do with the rig that has to do with manipulating or controlling something, like a game or election, to get a desired result. That rig is from a 17th century noun meaning "swindle." The rig in jury-rigged is a 15th century sailing term meaning "to fit out with rigging," with rigging being the lines and chains used in operating a sailing vessel. In the 18th century, if it was jury-rigged it was a boat:
La Couronne … bad bottoms, jury rigged.
— Morning Herald (London), 16 Aug. 1782"
Now back to EM parts