If a care insurance company offers you less than either bluebook or an estimate what options do you have?

First, who's insurance are you claiming against, yours or the other drivers? I ask because "liability" protects the other driver, not your vehicle. It's collision insurance that protects your own car. If you're claiming vs. the other person's liability, you probably have a slightly rougher road, since you have less connection to and leverage over someone else's insurance company. If you are claiming vs. your own collision, you're in better shape. I'll assume for the rest of the answer that you're claiming under your own policy, though even if you're filing vs. the other insurance, the same basic principal applies--insurance companies have to prove or establish to you why their offering the amount they are, if it's not the face value of the thing destroyed or damaged.

Insurance companies can't simply arbitrarily offer you whatever they want--typically (check your policy) they will pay either the repair cost or the current value of the vehicle, which is usually the blue book value. (Of course, they may be legitimate set-offs or reductions, such as the deductible if you're claiming vs. your own collision insurance.)

Your first step is to check your policy, so you know exactly what you can demand. Second step is to gather your evidence of what you think the claim is worth--repair estimates, the blue book value, and any other information that would establish the cost to fix or value of the vehicle. Third step is to ask the insurance company why they are offering you less than either repair or current value.

Once you know why they're offering less, you can evaluate their position. For example, if your car had excessive miles or pre-existing damange on it, it might be fair to offer less than blue book because a car with those conditions is worth less than blue book. If you disagree if the company's reasoning, then you can file a complaint with your state's department or division of insurance. You could also sue, though unless yours was a very valuable vehicle and the company is offering well under it's value, it is likely that a lawsuit would cost you more than you'd gain.