Always start with the execution plan. Look for full table scans and missing indexes first - they are the most common culprits and the easiest to fix.
Strong answers describe a systematic approach: using EXPLAIN or query plans, identifying full table scans, checking index usage, evaluating join strategies, considering query rewriting, and testing improvements. Best candidates discuss the trade-offs between query performance and readability, and when to denormalise for performance.
Practical SQL skill. Analysts who can only write queries but not optimise them become bottlenecks as data grows. Ask for a specific before/after example to gauge real experience.