"Ideas on" vs. "ideas for"

Solution 1:

Cambridge Dictionary Online suggests these proposition for "idea":

If you have any ideas for what I could buy Jack, let me know.

That's when I first had the idea of start ing (= planned to start) my own business.

And about your example:

I'd like some ideas for how to improve my team.

sounds odd and wrong. After "for" a ing-clause is used and after "on" a relative-clause.

Ideas on what to eat to night

Ideas on how to learn

Ideas on where to go

And

Ideas for eating

Ideas for living

Solution 2:

When you have some "ideas on how to improve my team," you have ideas relating to ideas on improving the team.

When you have "ideas for improving my team," you have ideas which specifically supports the team. For example, when you say

I am for peace-making

you are obviously supporting peace-making. In the same way, using "for" in ideas on improving the team means you support improving the team while using "on" doesn't necessarily mean so. It's all connotation and subconscious language use and effects.