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.