In the context of fashion, boring would be an antonym for flashy, and a "boring tie" would be one unlikely to catch someone's attention.
Collins defines flashy as "brilliant and dazzling"; Macmillan defines flashy as "very bright and colorful".
So, if these were the ties available in the closet:
I would say that (especially with a white dress shirt):
- a is a bright necktie
- d is a loud or ostentatious necktie
- c and e are plaid neckties, but they might also be called unusual neckties
and also – especially in comparison with the others:
- b could be called a boring necktie, or the most boring of the lot.
In this fashion context, boring could be a synonym for understated, although I'd probably consider understated to have more positive, professional overtones, and boring to be more negative, and perhaps a somewhat slangier term. If I thought someone was dressed too flashy, I might say:
Why don't you try a more understated tie?
whereas if I thought their outfit looked too plain for the occasion, I might say:
Why don't you try a less boring tie?
So, in a fashion sense, I might say:
Lady Gaga is not exactly known for wearing boring clothes.
as a way of saying:
Lady Gaga is known for her outlandish outfits.
With all that said, I think it's important to back up what mcalex mentioned in his answer. The author is doing some character development, and I very much doubt that Mr. Dursley's closet is filled with ties like the ones I've depicted above. Instead, I would guess that all of his ties are rather mundane, but, on this boring day, Mr. Dursley picked out the most boring tie out of all his boring neckties.