I believe many of these so called “beauty and the beast” personality traits are a matter of perception. If not the traits themselves, then one’s perceptions of another’s actions.
For example, if I have a friend who calls me or e-mails me often, maybe a touch too often, but I really care about this person, I’ll look indulgently on my friend’s efforts and think “How sweet! This person really, really cares about my well being, enjoys my company, etc, etc.” 
However, if a person I don’t like as much, or even *the same friend above who I happen to be annoyed at *, says and does the exact same things, I’ll probably think “this person is too clingy, needs too much attention, isn’t giving me enough space, blah, blah” (ad nauseum).:mad:
Of course, if I’m a fair minded person, and am as charitable of others’ foibles as I hope other people are of mine, then I’ll come around soon enough, and not draw unflattering conclusions because I happen to be in a pissy mood. Or maybe I won’t, in which case if the person is a true friend, I’ll have thrown that person’s friendship away, and be the poorer for it. And you know what? I’ll deserve what I get. Maybe I’ll end up alone.
Put more concisely, if you want to look for good qualities in most people, you’ll find them. If you insist on seeing the worst, you’ll find that.
Up to you.
As for specific qualities I have (or look for in my friends, I’m a bit confused as to what the OP is asking for), I place an emphasis on loyalty, and trust. I consider this a beauty and the beast type of trait in that it can blind me to flaws in my friends (quite possibly in myself as well).
One example: I had a friend who I placed a HUGE amount of faith and trust in. In spite of her telling me that she lies to her parents, cheated on her boyfriend on several occasions (both before and after he gave her “permission”…long story) and did other things that someone who claims to place a great deal of importance on integrity and forthrightness really shouldn’t be doing. I wrote some of this off to youthful indiscretion, and made excuses for the rest. (In fairness, I should state that she had, and probably still has, a whole host of really great qualities.)
Looking back, I shouldn’t have been caught so off guard when she acted less than honourably towards me. I’m not ashamed that I trusted/had faith in her, but in retrospect, it’s how caught by surprise I was that bothers me. (Another thing that really hurt was when she told me one day, well into our friendship, that she didn’t trust me, even though she couldn’t come up with a single instance where I was dishonest to her, and despite her own behavior, which, to give her her due, she freely admitted.)
Another quality I have and/or look for is persistence. (Kind of an adjunct to loyalty) I believe you have to work at your relationships (some, admittedly, more than others). I don’t consider it a hardship, really. If a friendship is worthwhile, particularly if you’ve shared a lot, it’s darned well WORTH working for. But then, it’s a labor of love for me to do things for my friends. I guess other people’s mileage may vary, but my thinking is, if you’ve got a good thing going and you hit a bump in the road, you work things out–not just give up, or make a half assed effort, saying you’re annoyed with your friend but not really giving him or her an opportunity to redress the problem. (I’ve been on the other end of that one, and it sucks. Basically I asked what I could do to mend things, only to be told “I don’t know”. Followed, in due time, by “Good Bye”. And it doesn’t make me feel any better to know that the other person’s loss is as great as mine.)
Briefer takes to some previous comments:
Re being ignored: I know the feeling, all too well. But then, if you put up with it, you deserve it…or so I see it. Everyone gets ignored/blown off from time to time, but where we draw the line: what we accept or refuse to accept, is often a huge determinant in how often we get blown off. Personally, in my friendships, I’d rather risk being trying to be around too much than not enough, and giving the impression that I’m ignoring someone. But then, it’s up to your SO–or friend–to communicate with you as to what the right amount of contact is.
Re being a drama queen: Interestingly enough, someone called me that! (I forget whether they used the word “queen”) Given that I don’t start threads about myself, my personal situations, my birthday or post parties (Heck, this post probably discloses more about me than 90% of my other posts), and I am often ambivalent about attention, I had to kind of chuckle about that one. I suspect the other person was projecting her own personal traits on me.
Re considering what other people consider your vices, you consider your virtues, there’s actually a job interview question that’s similar to this, and “spun adroitly”, the response above is pretty close to the remark that you should make. (Basically, it’s turning a “vice” into a “semi-virtue”: “I expect a lot of myself and often a shade too much of other people”, etc.)
Finally, I agree that the ability to see all sides is a pretty good characteristic for a mediator to have. I’d almost venture to say it’s necessary.