There are a varied number of words in Greek translated as Willing(ly, ness, etc.), but while some are more about wishing (boulomai) and others more about passionate desire and want/eagerness (prothumos), the word that seems to touch the idea that we as Christians go willingly (as Christ did to the cross) is theló. This is a word I struggle with when translating Greek. Is it related to desire? Is it related towards faithfulness (intent)?
No matter which way you go with it, the idea of being willing in the sense that we are able to will something to happen or in the sense that we do things with a certain attitude doesn't seem to exist here. This changes in some ways the way that many currently interpret the word. It may have something to do with attitude, but it is not about going and doing things because it's what we WANT to do, or because it will lead to things we WISH would happen.
I think Willingness is very closely related to faithfulness, which means we don't always know where we're going when we're being willing to move, it means that those initial steps may be done sadly, begrudgingly, maybe even with a little anger, but yet we move. And we move not because we fear the outcome if we don't but because we are open to that which moves around us. That is the willing that says "take this cup from me" in the garden, yet still heals those who are bound to destroy you. It is the willing that doesn't take the escape routes that are available.
For me, the idea of willingness is so connected to the idea of faithfulness in scripture, that it is something we cannot do alone, but it is connected to the design of God. Thus willingness is: That which we cannont NOT do. It is: That which we must, because our wish for the outcome to come (whatever it may be and unknown to us) is too great to overcome.