Strictly speaking, he answer is that there is no motivation for the name "jet" in the context of viscosity solutions for second order fully nonlinear elliptic PDE, because it was initially introduced in the more basic framework of differential calculus/geometry.
Still one can wonder about when and where it was introduced. I suspected it was initially introduced in French and possibly by Bourbaki, and asked J-P. Serre in an email, and got the answer:
Il me semble que la notion de jet, et la terminologie
correspondante, sont dus à Ehresmann (mais pas
à Bourbaki), vers 1950. Je vous envoie ci-joint un article de 1961 qui
(It seems to be that the notion of jet, and the corresponding terminology, are due to Ehresmann (but not Bourbaki) around 1950. Here is an article from 1961 confirming this.)
The article is Une généralisation du calcul des jets et quelques prolongements généralisés de variétés différentiables (DOI link linking to ProjectEuclid, no paywall) by M. Kawaguchi. Its first two sentences are:
Le mot "calcul des jets" a fait son début dans l'article
 de Ch. Ehresmann. En 1951, Ch. Ehresmann s'est intéressé surtout aux
fondements de la geometrie differentielle.  Les prolongements d'une variété différentiable, Atti IV Congresso Unione mat.
Italiana, Taormina Ott, (1951), 1–9.
(The phrase "jet calculus" first appeared in the article  by Ch. Ehresmann. In 1951, Ch. Ehresmann was especially interested in foundations of differential geometry.)
I should add that in French "jet" has nothing to do with plane or engine.
"Jet" is the substantive of "jeter" = throw, launch. I can see the translations: spurt, jet, squirt, throw, spray. Also a derived phrase is "premier jet" = "first draft".