If lichen amyloidosis is so rare, how could five members of my family have it?

Hereditary: It is rare but there is thought to be a familiar form of the disease that leads to the deposition of protein (amyloid) in families with the men2a gene (leading to risk for medullary thyroid carcinoma, pheochromocytoma, and hyperparathyroidism).

Genetics: About 10% report autosomal dominance inheritance, therefore there are genetic factors which appears to apply to your family.