I thought I'd put a bit of a positive spin on this kind of topic. Rather than discuss why we dislike certain characters, I thought we should share why the positives about characters we don't care for.

Miki/Cure Berry of Fresh Precure. 
I respect that she's confident without being arrogant and insulting to others. In one episode she loses the important trinket of the series, and refuses to explain how it happened, because she felt like she would be making excuses. Much integrity in that one.

Mana Aida of Doki Doki Precure.
When the show was first airing I really liked Mana. Her big flaw was her inability to think about the feelings of others when she's being particularly gung-ho, which helped deflect some of the cries of Mary Sue. That moment in episode... 7 I believe when she counters the villains evil laugh and declares that if he can teleport back and forth between worlds, they'll just beat the stuffing out of him until he takes them back home.

Orihime Inoue of Bleach
A lot of people hate her because she's "weak". I can respect her status in the show to an extent, since she's the kind of character who doesn't have the same fighting/killing instinct as the other characters. She's not going to be as combat focused as the others, so she's bound to end up as the support/healing character.

Shinji Ikari of Neon Genesis Evangelion
I think the same applies here as does with Orihime, but the circumstances are vastly different. Shinji's life is a level of crap that no normal human can possibly understand. Just look at who his father is, what he has to fight on a regular basis and one can't help but give him SOME slack for being "whiny, weak and cowardly".

Kamijou Kyousuke - PMMM
The guy did lose his big dream without any input or any chance of redeeming it. In comes his friend everyday full of energy just to (indirectly) remind him that his dream turned to crap, and he just couldn't take it anymore. 
Post Hospital, He might have even felt awkward since he just blew a fuse in front of her, and Sayaka didn't exactly appear to make an effort to speak with him. He's pretty dense, how was he supposed to know that it was alright for him to make the first move. They were also both 14, that's a pretty awkward age for boy and girl friendships altogether.

Hibiki Houjou/Cure Melody


After episode 35, she and Kanade do seem to have mellowed out their dynamic, and I enjoy her being an older sister to Ako.

Rei Hino/Sailor Mars


Mostly because the only reason I hated her was because I read SM while suffering from low self esteem, and she reminded me of all of the aspects of my personality I didn't like. (bad explanation is bad...)

Kyouko Sakura


I understand more about why she acted the way she did after her backstory, also her death in episode 9 had me crying the whole time, and I vastly prefer her in the Different Story.