This is a very broad question, but I can answer it from my own background. I have worked as a scientist in a research institute (at NASA) and have a PhD and both of these required that I deal with this question in a very big way. I'll try to keep this brief :).
In its purest form, research should address unanswered questions and unsolved problems. The research then is to find answers and solutions. If you know the answer to every question, then you're stuck. But no one does. One good source of topics is the "Areas of Future Research" that usually occur at the end of most research papers.
Another source is your academic advisor or teachers. They should be senior enough to know what are the open (and interesting) questions in your field.
THis does not apply to most people, but another good source of research topics are funding agencies. Government and non-government organizations frequently send out announcements of funding for research topics. The National Science Foundation (NSF), the Defense Advanced Research Projects Agency (DARPA), NASA, NIH and NIMH are ones I have worked with in the past.
For someone trying to devise, say, a science fair project, it is harder because almost everything has been answered. Pick a topic that is of interest to you, because doing a project is a lot of work and if the topic is not interesting, keeping your motivation up willbecome taxing.
Research is all about exploring and probing new areas that have not been worked upon. A researcher can choose any field or area that interests him and then gather all the existing knowledge about it. This would help to strengthen the grip of the researcher over the topic. Once you know the existing areas, you can see where there is need for further work. The discrepancies in the research can serve as the basis of your new research where you can try to explore new dimensions and avenues and update the existing knowledge. This would serve as a source for your research problem. For further professional, assistance you can take help from qualified research writers.
A research work require the researcher to put in complete dedication in order to come up with something that is yet not discovered and is new for everyone and for this the first requirement is to find your domain of interest and hence choosing the topic that you would love to work upon. One good source for completing your research is deducting from the previous theories and hence finding its discrepancies and conclusions. Another source for a new researcher is to interview subject matter experts and take reviews from them. Rest there are many firms which are running in collaboration with expert and professional researchers to offer professional assistance and guidance.
Personal experience can also be included in the sources of research problems. In the course of our professional training and experiences at work,we feel uncomfortable with some certain things. We feel that they should be changed or worked on. This allows us to embark on solving the problems we feel uncomfortable with.
There would be many sources you will find while doing a research. But sometime, for particular ones, you may not find. It's all you have to put complete soul and dedication. At-least you get some clue. All the best.