Soulnames are a protection against sending, which can easily reveal all of its user to the recipient. Soulnames essentially divide the elves inner-most self away from the rest of their mind, making it inaccessible to anyone - sending or otherwise - that does not have its key, the user's soulname. This protection is only needed by Wolfriders: the rest of the elves wear their soulnames as their usename. (I think there's a line to this mention in the novel Captives of Blue Mountain: Cutter is surprised that Winnowill wears her soulname so openly, or somesuch?)
Wolfrider style: Redlance, Stonefist, Treestump
Normal: Tyleet (gift of healing), Leetah (healing light), Shenshen (bright bright), Rayek (child of the rocks)
'Normal' style names all have meanings in the elfin language. (Need to dg up the old Yahoo group and forum thread about it.)
Moonshade discovered her soulname when she Recognised Strongbow in the middle of archery practice.
Skywise discovered his during a fight with Cutter, in which they had a strange quasi-Recognition.
Dewshine had hers discovered by Tyldak, when they Recognised in the Blue Mountain.