AMT income happens when the option is exercised (vests and paid for), the difference between the value and the strike price is income at that time. The AMT cost basis is the value at that time ... or you can think of it as the strike price plus the amount of income.

At the same time, if it's an ISO option, there is no assessment of ordinary tax until the stock is disposed. If there's a merger and the proper forms are followed, you can be issued stock from the acquirer that retains the basis (the strike price) of the original shares. If the forms are not followed, the acquisition is a taxable disposition.

There's a credit for the difference between AMT tax and ordinary tax on ISOs, but it can take many years for that to fully work out, and you have to have paid the AMT in the meantime.

Early exercise with 83(b) at time of grant (or while the value hasn't changed) or exercise-and-sell make the taxes make the taxes simplest, but tax simplicity isn't always the best strategy.