A good cover design brief should include the following elements (along with any additional information you might consider important for the designer to know)
What your aims are for the book
What place it will occupy in the book publishing landscape (ie. subject matter, genre etc)
The kind of feel or mood you would like the design to inspire or provoke. Give examples of existing titles – as many as you want, and what you found compelling about them – or other non-book material that is heading in the right direction – a ‘mood board’ can be quite helpful
A rough idea of how you plan to market your book, and whether it will be mostly promoted online or via bookstores, and what kind of additional marketing materials will be needed (posters, graphics for social posts, email headers, banners etc)
Examples of type design or font combinations that might set the designer on the right path
Examples of colour combinations, or the dominant colour
The blurb and a reasonably detailed synopsis, even a couple of key scenes in the book if you want them to be the basis of the cover
Character descriptions if they are to feature on the cover
Many authors are content to leave everything to the designer, but at least a little bit of guidance can be extremely helpful and prevent wasted time and the designer creating iterations that are wildly off-track.
Be open to unexpected solutions – sometimes a designer will come up with a solution that you might not have considered and showcases your title in an interesting, marketable way.
If the first round of cover versions are not hitting the mark, be specific with your suggestions – the more the designer has to work with, the more chance they have of creating something memorable and useful
There is a post on the WorkingType blog that goes into some related detail.