I believe you have at least two problems with trying to translate "τοῦ λόγου τῆς ζωῆς" as "living message" in this context.

First, **grammatically**, "living" is a participle (used in the phrase "living message" as an adjective modifying message/word), and so properly speaking, that English word would translate the participle form of the Greek verb ζάω ("live," such as in Mt 16:16), not the noun ζωή ("life," such as here).

An additional grammatical issue is the basic function and use of the Greek article, as Wallace notes in *Greek Grammar Beyond the Basics* (p.209-10, italics his, bold is mine):

>a) At bottom, the article intrinsically has the ability to *conceptualize*. Or, as Rosén has put it, the article “has **the power of according nominal status** to any expression to which it is appended, and, by this token, of conveying the status of a concept to whatever ‘thing’ is denoted by that expression, for the reason that whatever is conceived by the mind—so it would appear—becomes a concept as a result of one’s faculty to call it by a name.” In other words, **the article is able to turn just about any part of speech into a noun and, therefore, a concept**. For example, “poor” expresses a quality, but the addition of an article turns it into an entity, “the poor.” It is this ability to conceptualize that seems to be the basic force of the article.
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>b) Does it ever do more than conceptualize? Of course. A distinction needs to be made between the essential force of the article and what it is most frequently used for. In terms of basic force, the article conceptualizes. In terms of predominant *function*, it is normally used to *identify* an object. That is to say, **it is used predominantly to stress the identity of an individual or class or quality**. There are a variety of ways in which the article stresses identity. For example, it may distinguish one entity (or class) from another, identify something as known or unique, point to something physically present, or simply point out. The identifying function of the article covers a multitude of uses.
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>c) The Greek article also serves a determining function at times—i.e., it definitizes. On the one hand, although it would be incorrect to say that the article’s basic function is to make something definite, on the other hand, whenever it is used, the term it modifies must of necessity be definite.

So this relates to the phrase "τοῦ λόγου τῆς ζωῆς" specifically in that the genitive article was included *before* the ζωῆς; that is, rather than "τοῦ λόγου ζωῆς" (which is perfectly valid Greek to also express "the word of life"), by including the τῆς before ζωῆς as well, it is *emphasizing* the abstract *concept* of life (which is already a noun, so does not specifically need "conceptualizing"), and functioning to *identity* that idea more particularly. Wallace comments on this as well (p.226):

>Abstract nouns by their very nature focus on a quality. However, when such a noun is articular, that quality is “tightened up,” as it were, defined more closely, distinguished from other notions. This usage is quite frequent (articular abstract nouns are far more frequent than anarthrous abstracts).

In short, the inclusion of the article argues even more strongly that converting the noun "life" to an adjective "living" is violating the grammatical signal of the article that the nominal concept (the noun) is what is being emphasized. Had the word "life" been the participle adjective "living" or the straight adjective "alive" (NT Greek I believe uses the participle exclusively to denote the adjective idea).

Second, **contextually**, the concept of "life" is what carries on in v.2. That "life" (as noun, not as adjective for λόγος) is immediately pointed to at the start of v.2 "καὶ ἡ ζωὴ" as the primary subject, which is specifically "τὴν ζωὴν τὴν αἰώνιον" (here, the article before αἰώνιον [which is an adjective, not a noun, that makes a big difference in this case] is simply denoting the second attributive position for the adjective [see Wallace, p.306], meaning "eternal life").

So contextually, it makes no sense to push the genitive form of the noun ζωή to be doing an adjective's role for λόγος (a supporting role), when that noun ζωή is then made the focus. Rather, the word λόγος is supporting the idea of the manifestation and declaration of the concept of *life*.

###Conclusion

There does not seem to be good justification to try to translate the phrase as "living message," and the consistency of translations as "the word of life" has strong support for keeping it so.