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.
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.
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 identify 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 as an adjective, "living," or the straight adjective, "alive" (NT Greek I believe uses the participle exclusively to denote the adjective idea), then "living word" would have increased significantly in likelihood as it would have been an adjective the article was attached to (see the next paragraph for an example).
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.
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.
On "Meaning" of "the word of life"
While the question here is about translation (between two choices), a comment asks about what the phrasing "means." The genitive of τοῦ λόγου is to accommodate the preposition περί, but the genitive of τῆς ζωῆς is conveying some other idea typical of the genitive case. The human author (who I do take to be John the Apostle, who also authored the Gospel of John) jumps right into this statement as though the audience is familiar with what he is speaking about. This implies that his Gospel is already known by the readers here, for in that Gospel he was clear that:
- The Logos "became flesh" (Jn 1:14)—a reference to the incarnation of Jesus Christ, the Son of God, Who is the Word spoken of in John 1.
- The Logos was "in the beginning" (Jn 1:1a, cf. 1 Jn 1:1a "That which was from the beginning... the Word of life")
- The Logos was "with God" (Jn 1:1b, cf. 1 Jn 1:2d "that eternal life [which is the life related to the Logos in "Word of life"] which was with the Father")
- The Logos "was God" (Jn 1:1c)
- Through the Logos, "all things were made" (Jn 1:3)—so all other things "exist" by the instrument of the Logos
- In the Logos "was life" (Jn 1:4a)
So that is some background on 1 Jn 1:1's "the Word of life," and that helps the reader see that the genitive phrase "of life" in relation to the Logos is probably one of these ideas (categories taken from Wallace):
- Genitive of Content (p.92): i.e. the Logos full of life, or the Logos containing life (not this fits exactly the Gospel's statement of "in Him [the Logos] was life" in Jn 1:4a). This would be saying that all "life" is contained in the Logos (which matches that all existence apart from God Himself is dependent upon the Logos and matches that "eternal life" is all dependent upon the Logos).
- Genitive of Apposition (Epexegetical Genitive, Genitive of Definition) (p.95): i.e. the Logos which is life. If this is so, then the idea is emphasizing how the Logos and life are essentially one and the same; the expression (the Logos) is life itself.
- Genitive of Purpose (p.100): i.e. the Logos destined for (or moving in the direction toward) life. This emphasizes the intended result of the Logos. While this is true (theologically), the category is rare (according to Wallace), so not as likely a choice.
- Genitive of Product (p.106): i.e. the Logos which produces life. This would also be true (theologically), but is also a rare category.
I lean toward #1 as the intended idea, but any of those four I think could be argued.
The primary meaning is pointing the reader back to the previously discussed idea of the Logos in John's Gospel, and how that Logos incarnated (Jesus Christ) is to be made manifest/declared/witnessed of/etc. in relation to eternal life, which life is so that "you also may have fellowship with us, and ... with the Father and with His Son Jesus Christ" (1 Jn 1:3).