The primary sources
Let us start with what the sources actually say they were given, and by whom. First, from Völuspá:
- Mind they not own,
reflection they had not,
no vision nor cover
or colour fine;
mind gave Odinn,
reflection gave Hænir,
vision gave Lódurr
and colour fine.
Then, Snorri, in Gylfaginning, chapter IX:
When the sons of Borr were walking along the sea-strand, they found
two trees, and took up the trees and shaped men of them: the first
gave them spirit and life; the second, wit and feeling; the third,
form, speech, hearing, and sight. They gave them clothing and names:
the male was called Askr, and the female Embla, and of them was
mankind begotten, which received a dwelling-place under Midgard.
(comparing with Swedish translations I have, there are large variation; Lars Lönnroth translation of Völuspa lets the gods give breath, force, and blood and colour, respectively. There is less difference in Snorri's acccount, which is unsurprising as it is in prose; instead of "feeling", they are given the "ability to move")
As can be seen, the two sources that retell this even differs in almost every detail; Völuspa claims that it was Odin, as well as the relatively unknown Gods Hænir and Lódurr, while Snorri says that it was the sons of Borr, which he earlier had identified as Odin, Vili, and Vé. Vili and Vé are also almost unknown quantities.
Who are these gods?
Hænir also appears as the hostage sent by the Aesir to the Vanir at the end of the war between them, and he makes himself into a great chieftain, but he seems to be utterly reliant on the advice of the other hostage, Mimir. He also appears as one of the survivors after Ragnarök, where he selects something wooden that has to do with destiny (Lönnroth claims it is tools for divination, Steinsland that it is a new world tree). Finally, and perhaps most interesting, he also appears with Odin and Loki in the 10th century verse Haustlöng, which has made some commenters suggest that the otherwise unknown Lódurr might be Loki.
Vili and Vé are the little-known brothers of Odin, and they crop up here and there: they helped in slaying Ymir and the creation of the world from his corpse. Odin wou1d also, apparently, let his brothers rule in his stead when he was on his travels. Once he was gone so long that they married his wife...
There could have been more than one god to show that this was not something even the powerful Odin could achieve on his own (remember, the Norse gods are in fact rather feeble when compared to other pantheons); and three is a more "magic" number than two. However, there is also a suggestion made by Steinsland that this could have something to do with an earlier verse in Völuspá, where three giantesses enter the world of the Aesir and cause it to fall; apparently, there is structural parallelism to the point that the genders are wrong in this passage. As to why three giantesses, that is a good question, but it could be that they are actually the norns Urd, Verdandi and Skuld.
One thing seems clear, and that is that Odin as the most important god was also the one that actually gave them life. However, the other two gods, who also gave important things, are all decidedly vague and unknown (unless we take the hypothetical identification of Lodurr with Loki as fact), so that we can not really know why each gave what. This is reflected in the scholarly literature I consulted: even Clunies Ross, who is most interested in the matter, does not really devote energy on who gave what.
My take on this is that we should probably give more weight to the version in Völuspá. Even if the two versions might reflect two independent traditions, it seems more likely that someone introduced Odin's two brothers in a story where there originally were two other, little-known gods, than the other way around. That we have an early piece of scaldic poetry which puts Odin in a similar situation with at least one of these gods is also important. I would think that Hænir was an older, once important god that was almost, but not quite, forgotten.
Apart from translation linked in the text, I've consulted Lars Lönnroth's Den poetiska Eddan and Karl G. Johansson's Snorres Edda.
Gro Steinsland's Fornnordisk religion was used for most of the details on the different gods.
Britt-Mari Näsströms Nordiska gudinnor had a good introduction to Vili and Vé.
Margaret Clunies-Ross Prolonged Echoes has a lot to say about the creation of the world in general, and the ideology that is presented through it in particular.