So I had this idea for a Drow cleric who leaves the Underdark under orders from her patron deity Eilistraee to prove that there are good Drow who deserve a chance for trust and friendship. Is this something worth investing in or is it too close to Drizzt Do’Urden and am I part of the problem?

because you're playing a Cleric with optimism... no, you're fine. You aren't an edgelord soloboi missing the point of Drizz't's character.

I say go for it.

Thanks homie! Yeah, I’m thinking of making her like a Disney princess, but murder-y when the situation demands it. 

Yeah drow can be fun to play, not all of them have to be edgey edge lords of darkness.  Played a drow sorceress that never fit in well but was charasmatic enough to survive, was saved after a failed raid by her drow warparty by priestess from a temple of Sune and became a follower, was a very friendly and fun loving character after all the adjustment time and realization that the surface world didn't suck. (all pre campaign stuff)

Led to some drama later when her sisters found out she didn't die in combat and her mother was pissed one of her daughters converted and a whole heck of a lot of stuff.. BUT!

Whoa, that’s some good stuff right there! Did the whole “sunlight=bad” make her less fun to play?

 THe sunlight issue varies a lot depending on your campaign and your DM.

If you are doing a campaign that takes place entirely in the underdark there is little issue and your superior darkvision will be great in spotting the enemies before they see you (or at least at the same time as many creatures in the underdark have 120ft dark vision)

Similarly if most of the encounters are inside you have less of an issue than if you are travelling the open countryside for long periods of time

Some DMs rule direct sunlight not including sunlight passing through clouds so maybe 2/3 of the time you are outside you are not affected and a lot of the rest of the time you can find a shade of a tree

Some DMs might allow you to wear (or give you the opportunity of obtaining) tinted glasses, roughly a reverse of goggles of the night to remove the negative effects of sunlight sensitivity.

If you know sunlight is going to be an issue in your campaign playing a spellcaster that uses spells with a save rather than an attack will make a drow playable.

Eilistraee has been very active in recent years going so far as to appearing to her followers and her church is quite well received in places like Waterdeep (in Ed Greenwood's Death Masks there are even organized events to popularize the church with people "dressing as drows" and performing dance routines). Especially since the Moon Maiden has an alliance with Mystra and Lady Lareal Silverhand who rules Waterdeep as Open Lord is Mystra's Chosen.

So yeah, there is strong evidence to good drow presence in the Realms.

Thank you, Lathlaer! This is incredibly helpful info!

For some people any Drow PC is too close to Drizzt Do'Urden. If you and your DM can make it work then you're good.

I have a similar character but as a warlock with Eilistraee as a patron.

Daughter of the Drow by Elaine Cunningham (another Forgotten Realms novel) may give you some ideas as well. 

My brother has a 'character' in his back pocket for silly one-shots called Drizz Do'Over.  He's an in-universe Drizzt fanboy who adventures with his trusty swords Dullard and Dampgrip and who adventured with his pet cat, a black Cat he found in an alleyway once and fed until it followed him.  He's a ranger but he's not actually a beastmaster (think he's a Hunter, actually), and he is both perpetually cheerful and excited to be out adventuring and having to constantly remind himself - out loud, on the clock as it were - to try and act more like his inspiration.

It's amazing, in simultaneously the worst and best ways.

GIVE THIS MAN AN ANIMATED SERIES.