Do researchers study topics outside their field for fun?
Fundamentally, a researcher studying a different field "for fun" is in no other situation than anybody else studying "for fun". It's a hobby, just like reading, going to the theatre, or sports. Some engage in this hobby, some do not. Whether they "have time for that" is really the same question as whether they have time for any other hobbies - some do, some (especially on the tenure track) are maybe a bit crunched for time, and choose to use their little spare time on something a bit farther removed from their daily work grind.


  Is there any known case of a researcher who, by studying a topic outside his/her field for fun, had an insight for a novel approach or solution?


I am sure there is. Note that "having a an insight" is not a particularly high hoop to jump over. My students "have insights" all the time, the question is whether they are important, work well enough, and are generally more useful to pursue than the myriad other ideas they could use.

My gut feeling is that you rather meant whether somebody had a big breakthrough in a different field as a "hobbyist". I think this used to be more common in times past. I can't really think of a recent example (at least in the STEM fields), as plenty of time and resources (both of which a hobbyist definitionally does not have much of) is required for major breakthroughs.