Is it permissible for the the mother to refuse to forgive her ex-husband (the father of her children) in exchange for money, and is it valid for the children to pay this money to their mother?

Reconciling people is one of the greatest acts of worship. There is no harm in giving money to the mother so she may forgive the father. This is not a real sale or bartering, for forgiveness cannot be bought, and emotional harm cannot be compensated for, unless there was a material right of the mother over the father. Rather, this money is considered an act of righteousness (birr), kindness, and a mending of her feelings. It is also proof of your sincere desire for her to forgive your father, and an act of righteousness towards your father to absolve his responsibility. Therefore, offer righteousness and kindness to your mother, then appease her and encourage her to forgive, for in maintaining ties with her there is gain and well-being for you.