Step 1: Extract the pattern.
The single most useful line is the family history of both brain tumours and kidney tumours. When a hereditary cancer syndrome links the central nervous system to the kidney, one diagnosis dominates. So instead of going option-by-option first, anchor on the two-organ pairing.
Step 2: Map the brain-kidney pairing.
The autosomal-dominant syndrome defined by CNS/retinal haemangioblastomas together with renal cell carcinoma (and renal/pancreatic cysts, phaeochromocytoma) is Von Hippel-Lindau (VHL) syndrome, caused by mutation of the $\textit{VHL}$ tumour-suppressor gene on chromosome 3p. The patient's headache, confusion and brain tumour fit a posterior-fossa haemangioblastoma, and the kidney tumours in relatives fit the renal cell carcinoma component - a textbook VHL constellation.
Step 3: Exclude the distractors by their defining organ involvement.
• Neurofibromatosis - café-au-lait macules, neurofibromas, optic glioma / acoustic neuroma; the kidney is not its target.
• Li-Fraumeni - $\textit{TP53}$-driven; sarcomas, breast cancer, adrenocortical carcinoma, leukaemia - not a brain-kidney pairing.
• Churg-Strauss (eosinophilic granulomatosis with polyangiitis) - an asthma/eosinophilia vasculitis, not a familial tumour syndrome at all.
Step 4: Conclude.
Only VHL unites inherited brain and kidney tumours, matching every clue.
Final Answer: Option 3 - VHL syndrome.