Wordorigins.org

These two terms have different origins and different meanings, although they are becoming conflated in common usage.

Jerry-built, meaning shoddy construction, dates to 1869. From the 1869 Lonsdale Glossary:

Jerry-built, slightly, or unsubstantially built.

The origin of jerry-built is unknown. One assumes that it is somehow related to the name Jerry, but exactly how is not known.

Jury rig, while similar sounding, has a slightly different meaning, emphasizing the temporary nature of the solution and can imply an ingenious solution done with materials at hand. Jerry-built, on the other hand, is often used for a permanent, but poorly built, construction and has no positive connotation.

The origin of jury rig is nautical and is taken from the term jury mast. A jury mast is a temporary mast erected to hold sail when the normal mast has been lost due to storm or battle. From Captain John Smith’s The Generall Historie of Virginia, New-England and the Summer Isles, penned in 1616:

We had reaccommodated her a Iury mast, and the rest, to returne for Plimouth.

It is commonly thought that jury mast is a clipped form of injury mast, but no evidence of this longer term has been found. This form of jury is etymologically unrelated to the jury that sits in judgment at a trial.

The term jury rig itself appears in 1788. From Thomas Newte’s A Tour in England and Scotland published in that year:

The ships to be jury rigged: that is, to have smaller masts, yards, and rigging, than would be required for actual service.

(Source: Oxford English Dictionary, 2nd Edition)