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:
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.