What does the word "deck" mean in "deck transformation"? What's the idea behind this name?
My Algebraic Topology lecturer (P.T. Johnstone) claimed that it indeed comes from the German "decken", and that "deck transformation" is a mistranslation. (So he instead says "covering translations".) Assuming this is true, then perhaps the best place to look for evidence is in the German literature, maybe starting with Weyl?
I first quote Dieudonné's History of algebraic topology, p. 296f.
Previous work mentioned by Dieudonné is Weyl's treatment for Riemann surfaces (in Die Idee der Riemannschen Fläche, 1913), while he speculates that Poincaré and Dehn were probably aware of that result. Moreover, Dieudonné mentions an article by Reidemeister, Fundamentalgruppe und Überlagerungsräume, Nachr. Ges. Wiss. Göttingen, 1928, 69-76 (which I was unable to locate). I'm linking to his article in the ICM-proceedings of 1928 instead. In this latter, Reidemeister does not mention the word Decktransformationen (he speaks of transitive permutations by the fundamental group, for example), and as far as I can tell neither does Weyl use the word Deck in his works.
However, Seifert and Threlfall explicitly speak of the Deckbewegungsgruppe. The three pieces of that word are deck(en) - to cover, Bewegung - movement and Gruppe - group.
Finally, the etymology of the word decken. It has ancient Germanic roots and is the same as "deck" in English. I'm quoting the Oxford American Dictionary concerning that:
As Zhen Lin points out in the other answer, it seems quite likely that the word Decktransformation was simply taken from there, since Seifert and Threlfall was a long time standard reference in topology. I haven't checked later sources.