I told him I wouldn't give him any more money, but then I weakened.

weakened 是动词 weaken 的过去式，weaken 由形容词 weak（弱的）加上动词后缀 -en 构成，意思是"使变弱、减弱"。在这里 weakened 表示"意志变得不坚定"，即"动摇"的意思。