Usually, a person starts out with a single primary research interest as a PhD student (closely related to that of their adviser). But then as they move on through the assistant professor stage, they discover new research interests. Do they usually discover new research interests through collaborations/discussions with professors at their university or through conferences? Or do they discover new research interests through what can earn them funding (through the NSF and other agencies?) Especially since any topic could theoretically be interesting to them? (even though they might find some topics more interesting than other topics).
Is it more often that they do what they're most interested in, or that they do what they obtain funding for (and find what they get funding for interesting enough)? And can they sometimes even discover new interests through their PhD students?
Here are a couple ways that have resulted in new research directions for me:
Reading a paper/book. Reading a paper/book can inspire new ideas. You may find that a paper is missing something, or you think of a way of doing it better, or perhaps you can combine the ideas in the paper with something you already know.
Obtaining funding. Often funding is obtained in groups, generally to inspire collaboration and cross-fertilization of ideas. As a result, one is forced to venture into new territory.
Talking to a colleague (generally from other institute). Simply chatting with someone at a conference or when visiting another institute can inspire new research directions. They may have a problem; you may have the tools to solve it.
Changing job. This brings you in contact with new people who have new problems. Collaborations my result from conversations in the coffee room, or by going to research discussions within the new department.
Following up on something a student has done. You may set a masters student, for example, to look at something you find interesting, but have not yet had time to explore in depth. The student may come up with something useful, but then leaves.
Some points (I'm no Prof., but which research topics might "pay out" (wether citations, patents, fame) is imo rather a matter of common scientific sense):
It may be cheap/easy/time-efficient based on the current lab hardware and expertise/knowledge of the group to "dive" into a additional topic. So I would call this economical reasoning to exploit new research opportunities similar to funding reasons.
Publish or perish. Which research fields are trendy or will become trendy and offer high chance to publish and get citations. Visualisation tools for the publishing landscape or search engines might show you which fields are currently booming more or less. Or browse research forums like mathoverflow,citeulike,... to see what top and experienced scientist are currently interested in.
Finding a niche, which needs of course a really good overview on and understanding of a research field. Contrary to bullet point 2, focusing on a topic/problem no one else is interested in or thinks of may give you the chance to become a trendsetter, find a new effect/phenomenon and may earn you a lot "automatic" citations.
As much as possible, I'd encourage you to pick research topics because they really interest you, rather than because you think you can get funding for them. It's so much easier to bring passion and energy to your research when you love it (rather than when you've chosen it to get funding). That said here are a few more ways to develop new research interests.
Hear a talk at a conference.
Write a paper with a colleague who knows a different area than you.
Chase references for a paper you are reading or writing. I often learn quite a bit the first time that I write the introduction of a paper in an area that's new to me.
Changing funding opportunities. The unfortunate reality is that grants are often hard to come by, and it's not unusual for professors to tailor specific grant applications to the nature of the grants being offered. Depending on how unpopular the professor's previous research interests are at the time, this may result in a shift in research topics.
Single topic diverges into multiple topics. To describe this via example, at one time our lab was working on a neural imaging project related to human decision making. This research led to investigating decision-making on a larger scale, as we tried to create computational models. This led to a few papers detailing the applications of ontologies to behavioral and cognitive neuroscience. Simply by following the various aspects of the project we were able to investigate a wide range of topics.