I'm talking about incentive stock options (ISO), which is the most common thing to get. Non-statutory stock options (NSO) are entirely different and are not common for normal startup employees.
When you exercise ISOs, you are buying the stock at the strike price. It doesn't matter what the current market price of it is (other than for AMT). The cost basis is the strike price.
The "clock" for determine long-term vs short-term starts when you exercise, but it doesn't affect the cost basis. If you don't hold it long enough the gain may get taxed as short-term or ordinary income (usually no a difference except in a couple of states).