What is the proper way to ask for ideas for a research topic to a professor on a different institution that I have no connection with?
While it would be nice to imagine that we all do research for the betterment of mankind, the reality is that it is often transactional. This is because you’re asking for some stranger’s time and efforts, and they don’t see why they should give it to you. 

If you manage to show them that you are able to come up with non trivial results by yourself, or offer them some indication that they’ll get something out of it (a publication hopefully), you might get a chance. 

The reason that this is the way things are is because professors get approached with these requests all the time, and they have to prioritize them. First dibs go to their own inner circle of collaborators. Guess what circle random strangers are on? 

If professors responded seriously to each such request they’ll end up with no time, and because most of these requests end up as dead ends, they tend to be wary of allocating their efforts to high risk collaborations such as the one you’re offering. 

Good luck!

EDIT: if you really want to impress a professor, take their class and shine! Alternatively, open a literature review, and start doing the hard work of familiarizing yourself with the field. You’ll need to anyway if there’s gonna be a paper coming out of it. Also, write down your thoughts. Start a draft paper and write down your initial ideas. What is the problem? Why is it important? What is your proposed solution? Even if it’s a draft or rough or stupid to you, just write. Then you can send this to professors! This tells them two things: you’re serious about this and that you can formally write out your thoughts. It’ll also help you crystallize what exactly is it that you want to do.