If you are just asking what Telemachus uses to cover himself up when he goes to bed at night, the answer can be found at the very end of Book I. There, Telemachus goes up to bed and covers himself with a woolen fleece. Here is a quote showing this -- it's the last line of Book I.
But Telemachus as he lay covered with a woollen fleece kept thinking all night through of his intended voyage and of the counsel that Athena had given him.
In this book, Athena has appeared to Telemachus in disguise and has urged him to get rid of the suitors and to go looking for his father, Odysseus. Telemachus has berated the suitors and then has headed for bed. His old nursemaid, who loves him better than anyone else does, has led him up to bed, folded his clothes, and closed the door behind him (it's as if she's tucking him in for the night). He lies down under his woolen fleece but he doesn't fall asleep.