How to add/remove search fields from the search box dropdown in the navbar?
The quick answer is "no."

For some back-story, that feature was contributed (circa 4.3) by a volunteer from the community who was enough of a coder to be able to get it to work, but not really a CiviCRM architect. His implementation was done by hard-coding that list of fields directly into the template.

The result of his good-faith effort is that:


We have the feature (which IMO is a step up from previously not having it).
It's not flexible or extensible.


I've been thinking that it would be a great little project for someone to add a setting to the "display preferences" screen that would allow you to enable/disable fields from that list. Searchable custom fields could be included.

If anyone is interested in taking this on, let me know in the comments and I'm more than happy to help you get started.