(someone's) willingness to (do something)
When you say that you "are willing to ___", it means that you will say "yes" to doing it, but you don't really want to do it, or you're not happy about doing it.
For example, if your lazy grown-up child asks you for money, you can say:
I'm willing to lend you the money, but I expect to be paid back by the end of this year.
"Willingness" is a noun that we use to talk about that quality. It means how likely someone would be to do this difficult or unpleasant thing.
Her willingness to jump in and deal with all of the gritty details really impressed me.
My willingness to help was overpowered by my desire to go home.
When you use the word "willingness" in this way, it sounds intelligent and academic.