I’m wanting to play a drow in my new campaign and this is the first time I’ve really deeply read into the lore of drow and elves in general. 

I’m looking for unique ideas for drow player character concepts, something other than the usual “came to the surface because they were cast out” or “escaped to the surface because they hated drow society”. 

Anyone care to share their drow character’s backstory, or brainstorm some new ideas? 

One of my players is running a Drow Bard that is trying to find lore pertinent to demon summoning.  She once mentioned that it would be for vengeance against someone, but that has not had a chance to com up again.  The desire for lore that might not be accessible in her home can be a good reason to venture forth even if it is still incredibly selfish.  She is generally playing an evil character but not so much as to cause problems, but she is more than willing to kill if it will advance her cause.

Is there any desire that your character might have that would give them ample reason to team up with people they might otherwise look down upon as being weak surface dwellers?  Maybe the character wants to like other people but sees them as inferior, might even change their mind after interacting with them for a time.

I have a character I play every so often, a drow ranger (no scimitars or panthers XD) who grew up in a small village of Elistraee--worshipping drow on the surface. He's a Horizonwalker, because Eilistraee had made it known that her wish is that they guard the world from Abyssal incursions. That being said, he's still a practical person and does what it takes to survive. He usually ends up playing the Neutral a lot because of his connection with nature and his occasionally ruthless practicality.

One thing I would suggest for you is looking into the whole Drow pantheon. Even if you didn't want to go straight up 'good' and go with Elistraee, there are still several gods and goddesses to choose from. And even though Lolth might seem like the one that Drow are all about, there are several drow societies, both on the surface and in the Underdark, who serve the other members of the Drow pantheon.

If you have access to Mordenkainens Tome of Foes, it has a section dedicated to elves in general and a fairly large segment for the Drow and their pantheon.  If you do not have access to it I can give you access if you would like to read some of those sections for inspiration.  I think Jank has a good suggestion with that.

Your background could reference a problem in your drow society that requires contact with the outside world for some reason. And with that in mind, your character may have been sent to the surface not in exile, but for a specific purpose. The fact that your character may or may not be considered expendable could be the reason you were chosen over someone else.

"Pardon me, ma'am, but do you have a moment to talk about our lord and savior Lloth?"

I'm currently playing a surface drow cleric life domain serving Eilistraee.  Her backstory is that her parents fled the underdark when their house was found to be worshipping Eilistraee.  They were caught by their pursuers about 2 hours after reaching the surface, but were able to hide their child before being caught and dragged back into the underdark to whatever fate may await them, (the DM is taking sheer delight in deciding what to do with them for future storyline hooks for my character).  My character (the baby) was found by a group of Tabaxi that took the child with them and raised her.  This specific group of Tabaxi were followers of Eilistraee as well and when my character was old enough they sent her to the nearest Temple of Eilistraee for training as a Sword Dancer order Cleric.   She's been fun to play because while she's the "good" drow, she's never known the underdark (so the DM ruled she isn't affected by sunlight sensitivity but her darkvision is only 60 feet like other surface elves) and she thinks that she can bring other drow to the service of Eilistraee and the surface as members of the "goodly" races, since she has only ever heard stories of underdark drow and thinks they are just that...stories.  

I've been considering doing a drow Celestial warlock.  He's the elder boy of his house, and not terribly remarkable.  He frequently disappoints if not outright angers his mother.  He's perfectly mediocre.  He has a younger brother, though looking to move up in the house's hierarchy, and his mother is very pregnant.  If it's a boy, tradition states the third son is to be sacrificed to Lolth.

On a surface raid to gain slaves, the surface elves locate and attack the drow.  The drow quickly open a portal to retreat to the underdark.  Seeing his opportunity, the younger brother shoots him with his hand crossbow.  The sleep poison does its work and puts him fast asleep while the drow make their get away.  The surface elves leave him for dead.  When he wakes up, he is stranded on the surface.  He realizes what has happened.  Whether his brother just wanted to advance, or his mother wanted to potentially "upgrade" in the sons department is irrelevant.  He's stranded on the surface with no way to get home, and no future if he did.  He's got no choice but to bootstrap up and find his way on the surface.

You could always be a half-drow. They often share the prejudices of drow with fewer Underdark-related restrictions.

I came up with an idea for a incredibly devout follower of Selvetarm. She was so blessed by and adhered yo his teachings so well that she would just go on massive murder sprees, which was a problem for Drow society in the long run. But instead of fighting against her, they sent her on a holy crusade against the surface dwellers. But then she ended up loving the surface and now works as a gladiator in a really intense arena where she gets paid to follow her religion.

Drow can be really fun. My first 5e PC was a Drow wizard. A conjuror with a love for devils and elementals. They were mere stepping stones on his path to power. He worshipped Vhaeraun, god of male Drow, and sought the right to be equal to the various matrons of Drow society. The end goal of my character was to find a way to destroy the temple to Lolth in his home city. He did it. A series of spells to summon and bind a couple dozen abishai and a few elementals to destroy the whole place. After that he built a wizarding tower I put in a later game.

I always resisted playing drow, but my current character is one, and he is fun to play. I needed an idea I liked that would explain how a non-evil drow wound up on the surface.

My drow was a low-caste street urchin rounded up for arcane experiments by powerful mages. He was the only one of 666 subjects to survive experimentation (high Con). The wizards had hoped to create the perfect arcane weapon; instead they produced a wild magic sorcerer. Drow society marked him as a dangerous deviant and slated him for sacrifice to Lolth. He attempted to escape, triggering a powerful wild magic surge that teleported him to the surface. I started play with the Charlatan background, to help him blend in to the campaign setting (where drow are considered myth) and even used his false identity with the party for a while until the secret got out...

I never bought into those two stereotypical tropes. The drow outcast...so, it just never occurred to them to move to another drow city?!? The good soul...so, apart from an unexplained and unexplainable innate goodness, exactly what made the character refuse and abhor their own societal norms?!?

Here's a character I've been mulling about for a while...born and raised as a typical drow, goes on a surface raid on an Eilistraeean community while still young and unproven. The raid is a disaster, some raiders flee, some are killed, others taken prisoner - the char would be among those taken prisoner - the Eilistaeeans convert the char or at least show them there are other ways of life, years pass and the character eventually is set free and becomes a member of the community; at this point the char is an adult and has lived for a good long while on the surface, embracing a different morality but maybe retaining some quirks from his early imprinting (more reverence for females then males, still a slight feeling of superiority over other races, etc), but going from Chaotic or Neutral Evil to a more moderate Chaotic Neutral.

Eilistraee may be substituted with any other god or goddess, so long as the community believes that even a drow might be redeemed.

In FR lore you also have the underdark city of Sshamath that has overthrown the matriarchs and Lolth worship. It is still evil to neutral in orientation but led by male Drow magicians and warriors. Further one of the mages was able to cast a high magic ritual that converted undemon tanned Drow in the area back into the original pre retreat dark elves ( so brown to black hair and brown to dark brown skin. Further, Khelbin brought back an ancient Meyeritarian city that included many dark elves  originally. Most of these used a high elven ritual to combine their minds, bodies and souls into a different form - Sharns. This happened right before the spellplague so you could also be from a Sharn that has chosen to split becoming its original 3 Drow/dark elves.

Some years back I ran a perky goth drow necromancer who's backstory was that she was from a small drow surface community that traded produce to Underdark merchants.  They weren't really part of either stereotype, they were just ordinary people who'd found a way to make a living.

I have a concept for a drow that is a bit of a drider but is not. that sentance is a bit confusing but since i am a dm i have the concept of a drow i named Charlotte and she would have spider legs coming out here back with webs around the legs making it look like wings. she also i was planning ot make her a sorcerer or a warlock. and she would be of course a crusader for lolth. i plan to have my players meet here in soon. she also would look like a flapper as some called them (for those who don't know a flapper is just female who would dress in a way that during the time was consider suprising.) but alot of this concpets are kinda cool