Your inverter isn't likely to be able to handle that. I don't know what a hair straightener would use, but I would be very surprised if it was LESS than 500W.
Now, yes, it can be done. Those people with 12-15" subs in the back of their car are pulling 1000+ watts easy. But they do it by running a BIG fat wire from the battery, and sometimes have large capacitors involved as well (you wouldn't need caps for an iron). Also for the really impressive setups they use a much larger alternator to supply more power to/recharge the battery from the engine better than a stock one.
So, what I'm saying is yes, you can run a lot of power off your car, but a regular cigarette lighter inverter is probably not capable of it. They make big inverters, but you'll need to run better gauge wire than what is ran to the cigarette lighter. Assuming your wife doesn't have 5' of hair, she's not going to be draining your battery down enough to cause any problems if the car is running.
There is a possibility that your car has a "cigarette lighter" style plug that is solely for use as a 12V source for random stuff and inverters. In that case, it may run fairly hefty wire. Your car manual will probably tell you the max power you can draw from it. Keep in mind the DC to AC conversion is not 100% efficient. I don't know how efficient it is, but I'd guess ~70%.
EDIT: My girlfriend had a straightner with 1" wide elements. I just tried it on my Kill-a-watt meter and it pulled 168W from the wall. So it isn't near as many watts as I expected. I suppose if you have a 250W inverter, and if your car manual gives a max wattage inverter you should use (and its 250W or better) then you should be good to go. If you only have a 200W, I wouldn't feel comfortable with a 168W pull from it. My gf also had a brush curling iron thing and it only pulled 98W from the wall...