It's not about "identifying the (correct) noun"; it's about identifying the subject, because you're asking about the subject-verb agreement.
The number of teens using their cell phones while driving has doubled.
Here, the subject is shown in bold. In order to determine the number (singular or plural) of the subject, you need to figure out what the subject refers to. Does it refer to the number or teens?
You can easily figure out that it refers to the number unless you have no idea what the sentence means. In which case, you should be asking a different question on this sentence. But I'm pretty sure you do know what it means. So you know that the subject refers to the number.
Now, what the subject refers to is in the singular, so the verb must be in the singular as well. And that's called the subject-verb agreement.