How to overcome lack of ideas periods?

In my experience a major point is to engage with other's research. The most common occasion is to attend regularly seminars, and go to conferences. By listening to research talks, you will learn about tools which might apply to some of your questions, and about questions for which you might have the right tools.

Another possibility is simply to read other's research, but it is quite difficult to decide in what direction to read, while when you have a weekly seminar to attend you get in touch with ideas regularly without having to be proactive.

Maybe there is a subfield, with connexion with what you already know, where you heard things are happening and someone wrote a nice survey? Then read that and see if it leads you somewhere.

The existing answer is a good one, and essentially recommends that you look inside academia for research projects. Another way to approach the problem is to look outside academia. I find this approach to be more fruitful, and questions originating outside the discipline tend to have higher general impact, although it can sometimes be more challenging to find an outlet or a relevant research program to tie into.

Stack Exchange is a great way to find ideas. Find the stack or stacks related to your field, and just start answering questions. When you find questions that don't have answers in the literature, think of a way you can answer that. If it could lead to a paper, then do that and your drought is over.

Another option, closely related to the recommendation to interact with other people's research, is to talk to smart, educated people outside your discipline. This works best if your research area is not something highly specialized. It doesn't take long, in most cases, for a smart person to ask a question for which there is no accepted answers. Follow that question until you find the answer and then publish it.