In Arnold-Chiari malformation, the lower part of the cerebellum protrudes from its normal location in the back of the head. The word cerebellum means little brain.
It is a separate, smaller part of the brain located at the base of the skull just above the cervical spine. It slides down into the cervical or neck portion of the spinal canal.
There may be no symptoms with this condition, so the problem remains hidden. An MRI of the cervical (neck) spine would show the abnormal anatomy. But MRIs are expensive. They are not done routinely just to see if there's a problem unless there's a reason to do so.
It is not likely that the cousins of a child with a Chiari malformation (your children) would be affected as well. However, you should bring this piece of family history to your pediatrician's attention, and seek his or her advice.
Most likely, no testing will be recommended unless your child develops some kind of neurologic signs or symptoms.