It is true that you can't just rent an apartment in seattle, call it a day, and pay no income tax. CA will look closely at things like owning in CA, where bank accounts are, how many days you spend where (keep airline receipts), where you're registered to vote, where your kids/spouse live and attend school, etc. But it is possible to establish residency in a state tax haven and avoid that 12.3% CA income tax.
As always, if the amount of money is material, see a tax attorney.
Oh, apparently for NSOs. That is not true for ISOs, which the vast majority of startup employees will be granted. see Publication 1004 from the CA Franchise Tax Board, discussing ISOs [1]
Nonresident of California on Date of Stock Sale
Qualifying Disposition
If you exercise an incentive stock option while a California resident or a
nonresident and later sell the stock in a qualifying disposition while a
nonresident, the income is characterized as income from the sale or
disposition of intangible personal property having a source in your state of
residence at the time you sold the stock. Accordingly, you are not subject
to income tax by California even though the services that gave rise to the
grant may have been performed in this state. [pg 10]
For ISOs, you're only subject to AMT on the spread between fmv and strike price at exercise. However, since there is (I think by law?) a 90-day exercise window after leaving employment, there's only 2 circumstances: (1) exercise while employed, and (2) exercise w/in 90 days after leaving employment. In both cases you're almost certain to be a resident of the state in which you were employed. Nothing I know of would require you to pay AMT to the state in which you were granted the options you exercised, but because it's unlikely for you to be a resident of a different state, it's a less interesting question.
Also, if your company takes off like a rocket, it's very much in your interest to not procrastinate on exercising vested options, because you want to minimize the AMT-taxed spread between FMV and strike. That increases the likelihood you're a resident of the state in which you were employed when you exercise.
If you were employed in California when you were granted the options, you are going to pay CA taxes on them during exercise.