What if the research options available are not something you're too interested in? I had a professor tell me I could work with him, but he said I had to work on what he has decided. That doesn't seem very attractive to me.
With that perspective, it doesn't matter much on what specific problem you start your research. Instead, find out which research groups are doing amazing work in the area you would like to work. And apply there.
To identify these groups, read a lot of papers in your favorite research area.