I'm looking for a drama where one of the leads has something that makes them completely different from other characters in dramas. As for the criteria that makes them unique is entirely up to you, however I would prefer dramas without a love triangle.

New tales of a gisaeng - for the start, these seems like a cliche romance but when you watch it, its one of the unique ones. The characters arent uniqe, its more of the plot and narrative that is very unique and different. Female lead and male lead meet unexpectedly and they feel attraction for each other, but some unforseen circumstances lead them to miss out or cause big misunderstanding mainly due to the male lead. They do get together and are very perfect for each other until the guy unceremoniously dump the girl. And the woeing of the girl after the break up is one of the best unique loce romance ever cause she has become a gisaeng.

49 days - its a nice romance though there is love triangle in it, where in the female lead found out a little late that that the man she is in love with is a a fake prince charming who turns evil, while the cold jerk who seemed to hate her is the one man who loved her for the longest time

ugly alert - a reversal in the usual kdrama trope, here the female lead is a cold heiress who seemed to be such a bully while the male lead is the always happy and positive candy-type who supports his fa)mily even if it means going to jail for them. Not much of a love triangle, other than the villain who wants to get the girl, but for overall pf the drama they just ignore him.

lovers in prague
- i love how uniquely stalker-ish and upfront the female lead id in this drama. She even sometimes gets embarrassed at her own actions but she barrels through the embarrassment to confess and woe the guy she likes/loves
Ho Goo's Love - male lead is not the popular-type of guy, but he has a crush on a very popular girl since they were young. I loved the progression of the story and it is a very unique romance drama

The Lover - this is one of the most unique/realistic/hilarious dramas I've ever seen. The couples are all very interesting to watch and the situations they go through are very funny
LoveO2O - no crying, barely any misunderstandings, they don't even flinch if either one gets accused by others, ppl tried to separate them but they just didn't care much about those other ppl, no evil parents trying to separate them, no surprise Ex gf/bf from the past.

I call it unique because it's different from your usual asian drama romance (starts out good - misunderstanding - lots of crying - get back together - parents against the relationship - Ex/other party interested - etc)

Just a fair warning, their romance is NOT full of drama, so you might not like it. I understand most ppl watch Asian dramas for the romance/couple drama xD
Oh My Venus and Love O2O- Unlike the many k-dramas, the lead couples don't get into fights, don't cause or believe misunderstandings, and don't easily leave each other for ridiculous reason because they actually communicate with each other. 
I Got Married As A Job.... One of the most unique love doramas you'll ever see.
Bad Couple - It's the lead female pursuing the guy initially, pretty funny and really well written.

Also other interesting story lines in the drama. The music was a little different in a good way too :)
Date - Koi to wa Donna Mono Kashira - when jobless  "educated idler" aims to marry woman with a stable income; they are both very nerdy. Personally i enjoyed this drama very, very much.

Densha Otoko - the male lead is an otaku and the effort he makes in order to woo the main female lead is so sweet is not a typical romance at all.

That Fool - this one is about a contract marriage, the female lead is an actress and in order to avoid a scandal she fakes a marriage with an ordinary guy (male lead) he works at a post office and he is very nice person, he's not your typical handsome lead.

Goblin - the male leas is a inmortal God. 

The Master's sun - the female lead can see ghosts so she lives scare and barely sleeps due to this situation, but one day she meets the male lead and she finds out that if she touches the male lead the ghosts disappear and she no longer sees them. So she decides she will remain near him in order to touch him when ghosts appear. 

Bright it on ghost - the female lead is a ghost