If someone's willing, he has the quality of willingness, which can vary from meaning "prepared," to "enthusiastic," like your little brother's eager willingness to help walk the dog.
If your troop of Girl Scouts has a willingness to try anything, it means they stay cheerful and ready, even if they're hiking through a prickly field of thistles. When something is done with willingness, there is no sense of resentment or hesitation. The word's origin lies in the Old English wyllan, "to wish, desire, or want."
n cheerful compliance
eagerness, forwardness, readiness, zeal
openness, receptiveness, receptivity
willingness or readiness to receive (especially impressions or ideas)
undivided commitment or unreserved enthusiasm