There is a lot of coaching opportunity here for both the project manager and the intern. For example, we might talk about clear communication throughout the project, and the importance of creating check-in points to assess progress against the goal. It might be important in this case for both the manager and the intern to have been proactive in making sure expectations and needs were being met, perhaps after 4 hours of work, or after an initial milestone was completed. But even with all that, if you're an intern or a new hire, I'd like to make the case for doing additional work, even if it's beyond what you expected and even what you were paid for.
Work for Free
There are times when working for free is a bad idea. Established professionals with a strong reputation for producing reliable results might not be doing themselves any favors in working for free. I might advise folks who own their own businesses to have tough conversations with clients about expectations and scope and at what point additional fees or invoices might be produced. And I'm in opposition to completely unpaid internships, where students are expected to work 100% for free, contributing to an organization's bottom line. But in a case like this, when you're early in your career and considering working extra hours on a project that took longer than expected? You have more to gain by doing some extra work unpaid than the guy paying you probably does.
First of all, you've probably taken on this internship or project because you are interested in building experience in your field. And because you're still in experience-building mode, you can't really afford to blow it on this one. It's really important to develop some small wins - completed projects - early on so that you have some successes to talk about that will get you even more experience, and then get you the job or role you're gunning for. So finish this project, even if it requires putting in a little extra time that you aren't being paid for. Or, offer to take on an extra project, unpaid, to build experience in an area that you need more skill and demonstrated success in. You're building career capital here.
The other reason to work for free is that you probably want a great reference from this project manager. You want to be known as the kind of worker who "takes initiative", "goes above and beyond" and "exceeds expectations". So make sure that your manager can say that about you by putting in a little extra effort when you can. Add an unexpected element that you think would benefit the project, even if it hasn't been requested or isn't budgeted. Take the time to finish something well even if it's outside of the scope of what you originally planned or thought.
Go the Extra Mile
No one likes to be taken advantage of, and there are times when you want to set your boundaries clearly. But when you're early in your career, building experience and reputation, going the extra mile and doing some extra work for free is only likely to help you. Be the person that others love to work with because you perform ahead of the curve.