Something fitting for the race and backstory. I often use names from NPCs from one of the games(usually Morrowind) as either inspiration or I simply use the name.
Based on my characters background I will take a name that is still "right" for x race, but could be closer to y culture. For example I currently play a Nord in Skyrim who grew up in a remote, almost unknown Priory of Stendarr in Cyrodiil that was relatively untouched by the Great War. His name is Sernir which has been used on a Nord NPC on Morrowind, but I feel it is closer to Imperial culture than nordic even if it is a nordic name.

Edit: ops wdited something meant for another thread in

I liike the concept of what my main character(Grymsul Selbar) wore in his travels throughout the three TES hames I have played not inlcuding modded armors.
Morrowind- Her Hands Armor with different clothes from time to time.
Oblivion- Dwarven Armor with various helmets interchangeably. Never the dwaven helmet...
Skyrim- Dragonbone minus the helm with a circlet. http://cloud.steampowered.com/ugc/648747634021053862/93A01B778E521DC93C84EC790258AA4C9CABD440/

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I'm Japanese.. so mostly asian names come to mind.

My 1st game was Morrowind, and I use Sera (my real name, and since its unquie)
So when i got to Balmora, and someone went Mos Sera.. I just kept useing it for my Main Character.

Misaki, Hitomi, Makie, Manami, few others.

Most Khajit get named Durandal, or Something evil sounding (since there my assassians or theives half time.

Do you mainly play Mer or Men?

I use the random name generator by http://tamriel-rebuilt.org/?p=modding_data/ngen. Sadly, the Website is often non-responsive, likely because I'm not the only one using that name generator. I'll generate random names until I spot one I like; sometimes, because it slightly resembles my name.

I'll usually start out with an idea of a personality, and some sense of the character's general mores and perspective on the Empire, and towards the various major cultures, which is where my background knowledge of the setting comes into it. I don't want to overdefine a character at the outset; that tends to get in the way, somewhat as it does in roleplaying with other players. I tend to feel the real object of the game is to construct a coherent narrative, and I can't do that if I make too many presumptions about what the options will be.

In my first play-through, I was a Dunmer mage, named Bedave Selas. I tried to avoid spoilers, at least at first, so I didn't realize that there was actually a fair amount about Dunmer in Skyrim. It worked out rather nicely, as the College of Winterhold was a natural place for a Dunmer adventurer in Skyrim. My current character is a Nord warrior named Braste, who went straight to join the Legion. That character follows a theme I like to follow in roleplaying characters: someone who is straightforward and level-headed, faced with a world built on dream-logic.

Funny thing I've noticed, by the way: sometimes you'll get a dialogue option in which you tell the other person that you already know about some subject, as in, "Yes, I've heard of the White-Gold Concordat." Choosing such an option means bypassing a chance to hear that particular character's point of view on the subject, which is the sort of thing I'm looking for. This means, as a lore fan, it works better to play a character who makes responses like, "Thalmor? Who are they?"

It varys each game.

Morrowind... it was hard to (rrr lets face it, they ugly)
So i played Wood elf (Love crossbows and deiese resist..lol) (I roleplayed each race evently)

Oblivion
Wood elves look so horrable.. I made a Imperial (love fact they short) Switched to Bretons (face's fit more i guess..)

Skyrim.. its Toss up.
Hair designs.. lack. (I use one style, Short Hair) so its liimited me to point of.. being stuck as Berton and Imp... (I have a Wood elf and Khajitt to) I have pretty much avioded Altmers since they are "High" of themselves..
Thou I have to say... What ever game it is.
My charcters tend to look the same. (Ether white hair, or short hair..)

Sometimes, yes. Played Oblivion once with a Breton called Manus Rostorard. He was related to that Rostorard who's on Morrowind.

On Morrowind, I also started a Nord named Haggard Dark-Heart once without knowing there's a Dark-Heart in Oblivion. Later I decided that they're related, of course.

Also my first Skyrim character was Thorir Iron-Hand, and there's an Iron-Hand in Skyrim, so I guess they could be related.

Elvar Saryoni was the Dunmer with whom I did the Temple questline in Morrowind. Older brother of the real Patriarch Tholer Saryoni and the real writer of Saryoni's Sermons, he was blackmailed by his brother, cast out of the Temple and banished from Morrowind. He returned to have his revenge, though, and put Temple back on "right" track.

Poetius Gabinia, a Skyrim character. Imperial, Legionaire and a Bard. Related to Morrowind's only Imperial Bard, Maesa Gabinia.

Ma'Kaberr was the name of the Khajiit I once played. He was the main character of the book Trap.

Some other of my favorite names and characters ever without binds to lore:

Nelacayne (Altmer male, my only demigod character in Morrowind, has done almost everything. Main job: Archmagister of House Telvanni.)
Sinerinde (Altmer male on Skyrim, Arch-Mage and a necromancer to come who will retreat from the society he's currently living in. This he's a great man and related to Mankar Camoran.)
Hides-Swift-Strikes (Argonian male, my only demigod character in Skyrim. Leader of most of the guilds, Dragonslayer, etc. Being the Dark Brotherhood Listener is his #1 duty.)
Shade-in-Shadow (Argonian male, a Morag Tong assassin in Morrowind who also is a man-on-a-mission to free all the slaves in Vvardenfell.)
Levianor (Bosmer female, the first Morrowind character that went through the MQ.)
Jahn Ronald Reagan Scriebelle (Breton male. Never actually played him. Just invented the name for an article.)
Halvar Strong-Arm (Nord male, a Skyrim character I'm doing the Companions, Stormcloaks and MQ with.)

I usually play as an Imperial battlemage-styled character with the last name of Caed.

Morrowind - I play as Augustus Caed, the father of my Oblivion character. He serves the Legion, and is pretty well-known as a noble person.

Oblivion - I usually play as Jaekob Caed I, the son of Augustus Caed. Jaekob I is an Imperial battlemage as well, and he serves as a Blade and part of the Fighter's Guild. He (for the sake of roleplaying) gains several land grants and a financial grant as an additional bonus to saving Cyrodiil, and he also has a room in the Imperial Palace to live in during his stays in the Imperial City. Somewhere along the line, he finds a wife and has children. He becomes the great-grandfather of Jaekob Caed II, my Skyrim character.

Skyrim - I have quite a few Skyrim characters, but my primary one is Jaekob Caed II. He, as I said, is the great-grandson of Jaekob Caed I, Champion of Cyrodiil, and the great-great-grandson of Augustus Caed, the Legion officer. Jaekob II started off somewhat of a rebel. He had been trying to get into Skyrim, because his parents wanted him to join the Legion in Cyrodiil, but he didn't want to even be a part of it. That's when he was caught by the Legion and taken to his execution, of course which was interrupted by Alduin. Jaekob II goes with Hadvar, just because he seemed nice enough. He decides, after seeing the dragon attacking the tower near Whiterun and being summoned by the Greybeards that he'd fulfill his destiny as Dragonborn. As time went by, learning more and more dragon shouts, meeting Paarthurnax, and just learning about being the Dragonborn, that his duty to Skyrim is to maintain Imperial rule at this time, and help end the Stormcloaks' rebellion. So, he joins with the Legion, the ones whom he never wanted to associate with, just to help maintain balance. After he defeats Alduin, he usurps Shor's throne in Sovngarde (and yes, I realize this next part may not be lore-friendly, but it's pretty cool as a roleplaying plot ), and he absorbs a lot of Shor's power, making him like an avatar of Shor. (Forgive me if I'm confused as to what a Shezzarine is, because it just sounds better to say "avatar of Shor" ). He splits his persona and spirit between Skyrim and Sovngarde - leaving his spirit in the Hall of Valor and his persona (or physical body) in Skyrim. He helps the cities around Skyrim, and obtains Thaneship in many of the cities. He took up residence in Solitude in Proudspire Manor. He then met Ysolda, whom he married (I personally just did this part to get the benefits lol). Well, from there, I can't really tell much more, because the next game isn't out yet haha.

I usually play a character by the last name of Dren (yes, those Drens).

My Skyrim character is a Dunmer, named Voryn Dren, who is a nephew of the Champion of Cyrodiil, Norvalo Dren.
After becoming Count of the rebuilt Kvatch, the CoC had his sister, herself a niece of Orvas Dren, brought to Colovia. There, her son Voryn Dren was born in 4E150.

He grew up in relative splendor as he was, due to his relation to the Count, himself a noble of Kvatch. Tutored by many great and wise men of Cyrodiil, amongst them Janus Hassildor, Count of Skingrad and a select cadre of Mages Guild advisors, he was well on his way of becoming the Count himself some day when his uncle passed. But, during the confusion of the Great war when he was just 21 and his uncle mysteriously vanished, he and his family were forcefully relocated to the city of Cheydinhall near the Morrowind border.

Several years later (in approx. 4E176), just after the signing of the White Gold Concordat, young Voryn grew restless and decided to travel Tamriel. Still bitter at the Colovians who had chased his family out of the Hold they once ruled, and bitter at the Thalmor for re-igniting elven hatred in the Imperial heartland, he crossed over into mainland Morrowind. There, he joined a band of roving traders. Defending their caravans and learning their trade he came to visit several of the sacked ruins of his once-proud homeland. distraught, he left the caravan and drifted from place to place - from Morrowind into Black Marsh, from there into the twin kingdoms of Elsweyr and back into Cyrodiil.

Rummaging through the ruins of Castle Kvatch, he found clues to a darker side of his uncle - his secret mantling of the prince of madness, Sheogorath and his ties to the Dark Brotherhood. Now, on the heels of his familys hidden history he bribed a shady Khajiti smuggler to take him to the -now destroyed- island of Vvardenfell where he visited the region of Sheogorad and then the Isle of Solstheim.

It was in 4E199 when he returned to Cheydinhal to find his mother on her deathbed. Bringing the affairs of the family in order and paying a visit to the now destroyed Brotherhood sanctuary, he decided to follow up on another clue of his uncle's past: The hidden Blades redoubt of Cloud Ruler Temple.
There, he found clues pointing him to Skyrim. Upon stocking up for the long journey he set out to cross the Jerall mountains on foot - after several weeks on the road, starving and delirious from the cold he came upon an Imperial patrol...

I typically use a character native to the province the installment is set for my canon MQ character.
Dunmer for Morrowind, Imperial for Oblivion etc.

In Skyrim, my MQ canon character is Kriger Ice Breaker.