When does coming up with an idea constitute sufficient contribution for authorship?
I will give an answer that applies to mathematics and maybe some other things. It isn't quite your second point, but close. Background first. 

Newcomers to maths think of it as a bunch of facts. You learn some of those facts in early schooling but don't get a wider view. If you study maths at a higher level you think of it as proving theorems. Some facts and a bit of logic let you derive other facts. But if that is as far as you go, then you really aren't yet a mathematician. A mathematician is a person with enough insight into the workings of the thing that they can propose things that might be theorems if they could only be proven. But this isn't, to a mathematician, just making random statements. There has to be a reason why something is suspected to be true. Things that are likely to be true. 

So, if the kid just randomly proposes a connection between bananas and leukemia, then it isn't worthy of authorship. But if a person with deep insight into the workings of the disease and the properties of bananas proposes it as a topic of study then, yes, they could/should certainly be a co-author of the study even if others do the actual proof of the concept. 

It is the insight into the likely connection that has value. 

Otherwise, you might just be sent off on a random quest without any hope of success. 

However, in mathematics, if an advisor suggests a problem to a student, having the insight that the student hasn't yet developed, it is still pretty rare to demand co-authorship unless there is more of a collaboration in the development. But this is just the custom. Certainly it is worthy of acknowledgement, and maybe even co-authorship. It isn't even necessary in this case that the advisor have an outline of a proof, or even any deep insight into how to develop it. It is just insight into a problem worth pursuing. 

But if it is, in mathematics, not an advisor-student relationship and one proposes an idea to another, I suspect that co-authorship would feel more natural, but probably with an acknowledgement section in the resulting paper that details the contributions. But in this case, also, it is more likely that the work would be developed cooperatively, so the authorship question would be obvious. 

Other fields than mathematics have different views on authorship, of course, but it is, even there, a question of insight. Among an infinite number of roads, someone proposes one as worthy of following. If you follow that you owe them something for putting you on the path. How you acknowledge it is a matter of custom that varies from field to field.