I was considering rolling up a Runekeeper alt in LOTRO, and I even have an alt name reserved on Allarond's server. The only issue is that the name in question is Lyriana. I've since gone on and used that name elsewhere, for my EQ2 Dirge.
I've reused names from time to time. Lyriana has also been a Blood Elf mage, and a minstrel in the LOTRO beta. Cheerydeth has been a Death Knight, a Witch Hunter, and a Rogue, though I came up with an explanation for that. Then again, I knew that Cheerydeth the first was not going to be with me forever, and Cheerydeth the second's career got cut short along with my Warhammer subscription. By contrast, Lyriana (the third?) has had a successful and ongoing career, and I think I would associate the name more closely with the dirge than any other character I put the name on.
How about it, readers? Do you reuse names between characters, or is it important to you that your characters have unique names? Voting's open in the sidebar for the next two weeks or so.