Options value is usually described in terms of two factors; intrinsic value and extrinsic value. Intrinsic value of the option, as you surmise, is zero. Extrinsic value, though, is sometimes called the "time premium" and represents that portion of the option's value related to the probability that the option *could* end up ITM.
Contemplate that:
-all things equal, the same contract, but in two different months, will have different values. (the later dated expiry will be worth more, because there is higher probability of ending up ITM)
-all things equal, the same contract, but with two different implied volatilities, will have different values (the option associated with higher volatility is more expensive, also because of higher probability of ending up ITM)
My $5 put will never be worth more than $5. That's fine, when I bought it for $0.50.
Outside of my professional commodities experience, I have never intentionally taken an option to exercise.