His name is usually taken to be a nod to his father's skill with a bow. You're right about the 'far' part, but the second part I think is different (the Ancient Greek word for man is
anēr, andros - the root being
andr):
tele means "far away" or "from afar" (think 'television' it sees things from afar).
The mach root comes from a word meaning "fight".
So "Telemachos" is a "far-fighter", i.e. a bowman. Odysseus is primarily famous as an archer (as opposed to Achilles who is primarily famous for fighting with a spear).
I think that etymology is pretty well accepted. It certainly was when I was a student. I'm not sure that there's a completely accepted explanation for how the name fits into his place in the epic. Here's my take: Telemachus must prove that he can grow up into as good a man as his father. He does that throughout the poem, as he goes from being a very immature teenager to eventually fighting alongside his father at the end of the poem. Before the trial of the bow (when all the suitors attempt to string Odysseus's bow and win Penelope), Telemachus tries to string his father's bow himself. Here's a translation of the scene I have in mind by Samuel Butler[1]. The 'he' refers to Telemachus:
> As he spoke he sprang from his seat, threw his crimson cloak from him, and took his sword from his shoulder. First he set the axes in a row, in a long groove which he had dug for them, and had made straight by line. Then he stamped the earth tight round them, and everyone was surprised when they saw him set up so orderly, though he had never seen anything of the kind before. This done, he went on to the pavement to make trial of the bow; thrice did he tug at it, trying with all his might to draw the string, and thrice he had to leave off, though he had hoped to string the bow and shoot through the iron. He was trying for the fourth time, and would have strung it had not Ulysses made a sign to check him in spite of all his eagerness. So he said:
> "Alas! I shall either be always feeble and of no prowess, or I am too young, and have not yet reached my full strength so as to be able to hold my own if any one attacks me. You others, therefore, who are stronger than I, make trial of the bow and get this contest settled."
In that scene, I've always thought that Telemachus proves he is capable of becoming his father (he might have strung the bow), but he is willing - as a good son should - to yield to his father while he is still alive. At the same time, he also proves he is a good liar - like his father - since the little speech he gives at the end helps to encourage the suitors and think he is nobody to fear. When the fighting begins, however, he proves otherwise and he helps his father kill the suitors.
[1]: http://classics.mit.edu/Homer/odyssey.21.xxi.html