Mentor was the old nobleman in Ithaca appointed by Odysseus to bring up his son Telemachus. Although the English sense of 'mentor' would be perfectly appropriate for one appointed to rear and educate another, the name appears to derive more directly from meno with its several meanings such as 'to stay', 'abide', 'await', 'expect'. Perhaps menetos (formed from this) is the origin, this word meaning 'patient', 'longsuffering'.
On the other hand, perhaps the men- root found in such words as mnemon, 'mindful' and mnaomae, 'to remember' (and Latin monitor) might be the origin, so that Mentor's name means something like 'adviser'. And menos, 'spirit' should also be considered.
The modern English sense of 'mentor' to mean 'experienced and trusted adviser' came not so much directly from the Ithacan nobleman's name as from the French, since in the romance Télémaque by the seventeenth-century writer Fénelon the part played by Mentor (as a counsellor) is much more prominent than in the original Greek classical stories. Compare Mnemon.