In Odyssey 15, why might it be significant that Helen gives Telemachus a robe to bestow upon his future bride?
This strikes me as a really interesting question. On one level, the gift of a robe from Helen seems to mark Telemachus as a young man who has now reached an age at which it may be appropriate for him to marry. So, acquisition of such a robe may be sort of a sign that Telemachus is "all grown up."
On another level, the gift creates a bond of guest-friendship between Telemachus, the future king of Ithaca, and Helen, the current queen of Sparta. Gifts of guest-friendship were like personal treaties between individuals, which could even carry over to subsequent generations in Homeric culture (compare the guest-friendship between Diomedes and Glaucus' fathers in Iliad 6).
Another possibility is that Helen might imagine Telemachus as a potential match for her own daughter, Hermione, although the mythic tradition will eventually link Hermione with Achilles' son Neoptolemus.