If a mother suffers from a thyroid problem, at what age might a child also get this problem?

Any age. We see autoimmune thyroid disease in children as young as babies and toddlers rarely. Congenital hypothyroidism occurs when a baby is born without a functioning thyroid, though usually not related to a parent's thyroid problem. Though the risk to acquire it is somewhat higher in kids whose parents have autoimmune thyroid disease, it may not happen at all, or it could happen at any age.
Hypothyroid. Find out if you have hashimotosthis disease causing your thyroid problem. This is an "autoimmune" disorder that has a strong genetic component. It is 7 more times likely in females than males. Typical ages to show up are adolescence, pregnancy, and menopause.