Should the mother's vow to pay money on behalf of her sick son in a specific shrine be fulfilled, or what should be done?

If the vow was made to the owner of the shrine, then it is shirk (polytheism) from which one must repent, and nothing is binding; because it is a vow made to other than Allah.

If the vow was made to Allah but for the purpose of the shrine’s upkeep, then it is a vow of disobedience that must not be fulfilled, and one must repent from it. No expiation is required according to the majority of scholars.

Based on this, there is no obligation to take your mother to the shrine, and it is not permissible to fulfill this vow.