Is there sin upon a mother who takes money from her husband without his knowledge to enroll their son in private lessons, given that the father refuses despite the son's need?

The fundamental principle is that a wife is entrusted with her husband's wealth, and it is not permissible for her to take from it without his permission, unless the husband refrains from providing the obligatory maintenance for her or their children. In such a case, it is permissible for her to take what is sufficient for them in a reasonable manner, as Hind bint Utbah did.

Therefore, if your son needs these lessons for his basic education, there is no sin upon you for taking from your husband's money without his knowledge to cover the costs of these lessons.