Well, let's see...
Generally I try to give my characters as cool of a name as I possibly can, which is pretty easy for me since, not to toot my own horn, but coming up with cool names isn't exactly a difficult task for me. I've played all the different races in Morrowind, taken most of them to Level 100 and one end of the game and back again, and so have developed a bit of a pattern whenever thinking of doing so again when I play these days. I have some that I've used multiple times as well, simply because of how much I like/liked the names.
Bosmer: For a male, "Rormank the Ridiculous". For a female, "Kameeva Vile". Rormank is typically a lot more powerful, and ignores the law to such an extent that I can't possibly finish the MQ with him, ever. At least not without cheating. Kameeva Vile comes from a long line of evil jerks, and really doesn't like it, so she jumps at the chance to save Tamriel from old Loincloth Man.
Altmer: For either gender, I use the rather androgynous-sounding Adoriene. At least, it sounds androgynous to me. You tell me someone's name is Adoriene, I will have no clue whether I'm about to meet a dude or a broad. Don't play as Altmers often though, since they get too-easily pwned by Ascended Sleepers, the Dagoths, and anything that I forget has Reflect.
Dunmer: For a female, "Xetliti LeVereo". For males, "Oruon Rassiddimmun". No real explanation for either one, or any commonality in character really. I like to name the males "Jiub" as well sometimes, even go so far as to make my character look like the original Jiub.
Nord: I only ever play as male Nords lately, and they're almost always "Hronfrim the Unstoppable". Because generally-speaking, my Nords are complete and utter war-machines.
Kahjiit: I usually name them after other people's pets, like "Riley", or "Cuddles". Can't help it, they're cat-people.
Redguard: For males, "Enambe War-Singer". For females, something similar. Haven't played as a Redguard in awhile, I find them to be very a "beginners" race lately.
Imperial: For either gender I go with something Roman-sounding typically, like with males I'm almost always "Resilius Demolishus".
Breton: For males, "Daran Turonus". For females, "Danica Turonus". Bretons are boring to me, and so they get boring names.
Argonian: This is where I get the most creative, and, generally, the most silly. And offensive. I can't list a good 80% of them, the censor will immolate my post, but I can tell you my favorite, and most-used: "Kills-You-In-Your-Sleep", a Fighter-Thief with a very bad attitude.
Orc: Ah, my favorite race. I have as many names for them as I do characters that I've made. Agronak, Hilmgrac, Alderak, Shorzigurg, Vorandbur, Jurdock, Grothis, Tyralninar, Deragbaknumbigarr, the list goes on and on. It also includes some names I plan on using in my book, so I can't afford to let them be known just yet.
I only ever play as males too. Females Orcs in Morrowind svck.
Oh, and as for what name I'm thinking of using when I first play Skyrim, I will probably play as an Argonian named something I can't post here.