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###Conclusion

Conclusion

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):

1. 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).
2. 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.
3. 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.
4. 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).

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):

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).

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):

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).