To clarify what was indicated by other commentators, you cannot get any equity in a non-profit organization, but a non-profit can own equity in a for-profit entity. So you can designate equity in a new company to go to the non-profit for their contribution of the idea.
Alternatively, you could offer the non-profit a royalty calculated on units sold . . . as is done with publishing (10 to 15%) or with licensed intellectual property such as a patent. Surely they understand that if they are only contributing the idea, they will get less than 50%, so don't hesitate to just ask them, "what percentage do you think is fair?" then negotiate from there, and if necessary, help them to understand that you are taking all the risk.
By the way, if the product/invention is one that is relevant to their non-profit work, they can invest in the development and can own the product in it's entirety, and instead they could pay you a royalty based on their sales. In that scenario, they may be able to approach their big donors and convince them to make donations to the project in order to develop the long term financial viability of the non-profit based on revenue from this project.