Although, unlike other myths in which a shared name with a plant and premature death, results in the rise of a plant bearing his name, that does not appear to be the case for Linos. However I do think that there is a very important relationship ongoing here that is related to the cult of Apollon and the production and manufacture of linon (or flax) that was used for making linen thread and cords, both useful for sails of ships, ropes, for making cloth (including funerary shrouds which make a particular sense in this association) and in regards to the strings of instruments.
As Apollon often makes, or is highly instrumental, in transitional myths going from one mode of lifestyle to another (taking the judgment of Orestes into account for instance in which a new, more just, legal system takes the place of the previous) it would probably not be surprising to see him have some relation to a material that would be, in many cases, a beneficial replacement to uses of wool and leather goods. As flax (or linon) doesn’t grow very well in Greek environment, being too dry overall with a terrain not hospitable for the maintenance of the plant, it is quite likely that linon was a product that came later into Hellas from its expansion and trade with Asia and Egypt. Keeping in mind that flax has flowers that bloom for barely a day before dying, being mythically tied to premature death in line with other tender flowers such as Hyakinth, it is quite certain that it would have been seen as a kind of noble plant and quite precious. The myth of Apollon’s son, exposed to die as an infant but found and reared by shepherds before dying a horrible death torn apart by wild dogs (not too unlike his contemporary and brother Orpheus who was torn apart by Maenads either) can certainly seem to be a stand-in to show the introduction and fostering of linen making linon among shepherding wool producing landscape. The associations with the arts of music (like his father) further reinforces his association with what is “civilized” (pure cloth linen versus pure raw wool may have been seen as a civilized upgrade and as a luxury imported probably was initially found more among the wealthy) and beautiful.
Here his death seems to be more of a sacrifice, which accounts for various myths in which he dies. As above I mentioned how he was torn apart by wild dogs, the result of which was penitence by the city by singing dirges for Linos yearly. Another is that Apollon sentenced him to death after boasting his skills as a musician and competing with Apollon in musical contest (for which a statue of him was erected at Mount Helion where the Muses dwell and where he had this contest). Out of the three versions though this seems to have less to do with the other two other than remarking simply on the superior quality of Linos. And then there is his death by Herakles over a bit of temper on the part of the Hero for his inability to master the elegant arts as a pupil of Linos (who himself was said to have taught music to both Orpheus and Museaos) and so meeting death by accident when being struck a bit too hard by the hero.
Of everything it is the song of Linos, the funerary dirge form which remembered his name as an ailinon. This remarks on the death of life that is sacrificed and became a popular form of lamentation. The associations with his playing on linen lyre strings (probably suggesting that he was the first to play on strings made of linen) and the construct of the burial shroud which an important part of the burial process (remember how Penelope spent years fending off her suitors with the important task of making the burial shroud of Odysseus) probably both work together in highlighting the importance of the plant that is subtly shown through the myths of Linus.
What is interesting to consider is how this relates to Apollon’s relationship with Athena who herself is a weaver of all manner of things, and himself as a god who produces and cultivates the finest goods. He is the herder who rears the beasts for fleecing just as he is the father of the linon (and sometimes its slayer…although wild dogs also can point to Artemis too) who presents a profitable and most beneficial produce to Hellas.
The fact that flax, especially as an oil, when consumed provides many health benefits was probably not lost upon them either and was utilized as much for healing benefits that would have been likewise under the providence of Apollon.