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Bruns/Herzog "Cohen-Macaulay-Rings" has a note in the notes for Chapter 1, saying roughly that after the influx of homological algebra into commutative ring theory, modules became popular objects (instead of ideals). They cite Gröbner's 1949 book "Moderne algebraische Geometrie" as the birthplace of "Vektormoduln", which are submodules of free modules. When did the term "module" (with its current definition) appear first, and why would the word "module" be chosen for this concept?

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Just one trivial note: Ideals are modules! When I learned this, I thought that it was a cool fact. –  Spice the Bird Oct 5 '11 at 20:13
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The site jeff560.tripod.com/s.html has answers to many of these types of questions. There is an entry for module. –  Chris Oct 5 '11 at 20:26

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From the website Chris Dionne mentioned in the comments:

MODULE. A JSTOR search found the English term in E. T. Bell’s “Successive Generalizations in the Theory of Numbers,” American Mathematical Monthly, 34, (1927), 55-75. Bell was describing the work of Dedekind, basing his account on Dedekind’s French article, “Sur la Théorie des Nombres entiers algébriques” (1877) Gesammelte mathematische Werke 3 pp. 262-298. Dedekind used the French word module to translate his German term Modul. Stillwell writes in the Introduction to his English translation, Theory of Algebraic Integers (1996, p. 5), “Dedekind presumably chose the name ‘module’ because a module M is something for which ‘congruence modulo M’ is meaningful.” Curiously le module had once before been translated into English but then it went into English as the MODULUS of a complex number. [John Aldrich]

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Hardy and Wright reference the advanced use of the term 'modulus' in An Introduction to the Theory of Numbers - not sure which edition this would have been, or the date. –  Mark Bennet Oct 5 '11 at 22:33

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