I don’t know about “a thing,” but it’s common to say “something.” It sounds natural (to me) to say “Have you got something against bow ties?” but not “Have you got a thing against bow ties?”
This. And the rest of the context determines (or at least suggests) whether the opinion is favorable or not.
Or a strong feeling.
Here are two examples of have a thing for in context from news media, one with more or less positive, and the other more or less negative, connotations:
[QUOTE=Tavis Smiley Show, May 26, 2006 – Interview with actor Chris Gardner]
Gardner: The bags (laughter).
Tavis: The bags, plural (laughter). Your shopping cart full of bags. Tell me what happens when you get a place and your son - go ahead. You tell the story.
Gardner: Well, the first thing I can say to you, I have a thing about bags to this day. I can not throw bags away.
Tavis: So you got a houseful of bags (laughter).
Gardner: I have a room in my house with nothing but bags in it, all right?
Tavis: Got some nice ones too, I’m sure (laughter). Neiman Marcus and Saks Fifth Avenue.
Gardner: No, no, but I can not throw bags away. . .
[/quote]
[QUOTE=CNN- AM
May 30, 2006 - Van Marsh on “The Power of Music” - Will I Am]
VAN-MARSH: One of Bruno’s musical idols, the American group the Black Eyed Peas. And like Bruno, members of the Peas came from rough backgrounds. (on-camera): This is a return trip to South Africa for the Black Eyed Peas. They performed here a few years ago but said that they were surprised to see so few black in the audience.
WILL.I.AM: Not that I have – not that I have a thing about performing in front of white people. I love Switzerland. I [just] didn’t think I was going to come to Switzerland in Africa.
[/quote]