On MathJobs, what does "Faculty contacts in our dept?" means?
This is a really really important question, and it's really important that the correct answer is option 2.  Most graduate students interpret it to mean 1, and this puts them at a disadvantage.  You should list any people in the department who might be interested in serving a postdoc mentor for you.  You should always list someone.

There are two main reasons for this:


On mathjobs you can run a search for all applicants who listed a given faculty member as a contact.  I always look at the applicants who list me as a reference.  Along with sending an email, this is the best way to assure that at least one person looks at your file.
When I served on the postdoc hiring committee I was responsible for looking through 20% of the applicants.  We're only going to hire a postdoc if there's a good research match for them in the department who is interested in mentoring them.  If I see a good file, one of the most important things is to make it easy for me to know which faculty member to mention the file to.  Sometimes I can work that out myself, but it's not always obvious.  Listing someone as faculty contact immediately indicates who I should email.