Step 1: The question keys on which polyposis syndrome carries cutaneous cysts. The answer is Gardner syndrome, an FAP subtype with conspicuous soft-tissue and bony lesions outside the gut.
Step 2: Remember the triad-plus of Gardner - bone (osteomas), connective tissue (desmoids, lipomas) and skin (epidermoid cysts), riding on top of the underlying adenomatous colonic polyps that carry near-certain malignant risk.
Step 3: Rule out the rest - Turcot pairs polyps with brain tumors, Peutz-Jeghers gives hamartomas plus lip and buccal pigment spots, and plain familial polyposis coli has no extra-colonic skin findings.
\[\boxed{\text{Gardner's syndrome}}\]