you'll still take home a large payout even after taxes have been deducted, i would not worry about that, would be more worried about when/if the company either goes public or gets acquired.
But they will be on the hook for an AMT bill in April which will likely be worth 3-10 years worth of their current annual salary. The tax question is by far the most important question for this individual. The question of losing the $X0,000 that they put into the options directly is a rounding error.
After exercising, they now own shares. The difference between the purchase price and the value of the shares is income, and OP needs to pay taxes on that income.
Note that OP isn't getting money, they are getting the shares, which they can hopefully sell in the future, either in an IPO or acquisition (or future internal round).
If they want to sell the shares to get money (to pay the taxes), there are tricky rules involved.