Exercising an option early if you can't sell the underlying stock being purchased is generally not advisable.
You're basically locking in the worst price you can possibly pay, plus you're losing the time value on your money (which is, admittedly fairly low right now, but still).
Let's say you have a strike price of $50. I get that you believe the stock to be worth more than $50. Let's assume that that's probably, but not certainly right.
Whether it's worth $51, $151, or $5,100 when your options are going to expire, you still get the profit of $1, $101, or $5,050 if you wait until expiration and exercise then. By exercising now, you're giving up two things:
If you buy it now, you get all the upside above your strike, but have all the downside below it. If you buy it later (at expiration), you still have all the upside above your strike, but no downside - in the (assumed to be unlikely) event that it's worth less than the strike you can simply do nothing, instead of having something you bought at the strike that's worth less now and taking that loss. By exercising early, you take on that loss risk, and give up the interest (or "carry" on the money you spend to exercise) for no additional updside.
It's possible that there are tax benefits, as other posters mention, but the odds that "starting the clock" for LTCG is worth as much as the "optionality", or loss protection, plus the "carry", or interest that you're giving up is fairly unlikely.