You will not need to do a "visa run."
"Visa run" is the term for a trip taken so that a person who's here in South Korea can obtain a visa at a Korean consulate in another country. Because you won't have a visa at all, you won't need to do a visa run. What you'll have to do is leave the country and then re-enter. You hold a US passport. American citizens are granted 90 days in Korea on a tourist visa, but other than a stamp in their passport, they're not issued any formal documentation. When leaving South Korea, the date on your entry stamp will be compared to that of your departure. If you haven't overstayed, you'll be fine to depart. Then, when re-entering the country, you'll receive another dated entry stamp and you will be permitted to remain in South Korea for another 3 months.
BUT, here are a few things that you ought to consider...
#1: Immigration officials are not idiots. They're going to see that you just left Korea very recently when you try to re-enter. Let's say that you took a ferry to Japan, stayed one night, and then returned... Or that you went to Hong Kong for a weekend, etc. They're going to see that and they're going to ask you why you're coming back. They'll know that you stayed for 3 months and then left for anywhere from 24 hours to a week or whatever the time frame might be.
What are you going to tell them? Because you ought to know that when entering South Korea on a tourist visa (which really ought to be called "visa waiver for travelers" because as I said, no "visa" is issued), you are not permitted to engage in commercial activities OF ANY KIND. It doesn't matter whether or not you're being paid. You are permitted to travel. You are not permitted to work, study, or do any of the things people need to obtain visas to do. And because you will have been in Korea for the previous 3 months, the immigration officials will certainly suspect that you have been doing some type of work. And they can bar you from re-entering at their discretion, so you ought to have a really solid story worked out in case you are asked to provide specific details. Now, that might be all for naught, and maybe the official you deal with won't give a toss and simply wave you right through. But if that doesn't happen, you could be in a bit of a pickle, so definitely decide what you're going to say in that situation.
#2: Where are you planning to stay for 3 months? If it's an unpaid internship, it's doubtful that they're going to be providing free accommodation. Do you plan to stay in hotels for 3 months, because that would be incredibly expensive. Hotel rooms are relatively cheap in Korea compared to other developed countries - if you look hard enough you can find some pretty good deals. But even with the average price for a hotel room being between ₩40,000 and ₩100,000 per night, it would still cost you about $3,500 if you were to stay in hotel rooms the entire time, and that's a low estimate.
Without a valid visa, you won't be able to rent a cheap room because people need to provide a specific set of documents to rent an accommodation in Korea and you won't have those. Not to mention the fact that you won't be able to open a bank account, so you're going to have to use a foreign card for everything, or use cash. That means that you won't be able to register a mobile phone either unless you purchase one from one of the kiosks at the airport and use it pay-as-you-go. Your American mobile phone will pick up any wireless internet that's available, but you'd have to use OTO or Skype to be able to make calls if you don't have access to a phone that's connected to the Korean mobile network. But back to accommodation, if you walk into a Korean hotel or motel and ask them to quote you a price for a week or a month's stay, it's going to raise eyebrows. Even goshiwons ask for Korean ID cards, so unless you find a very dodgy landlord who doesn't ask questions, you're going to have a nightmare trying to find someplace to sleep every night.
There are plenty of hostels that probably wouldn't ask questions either, and some of them are cheap as chips. You could book a bunk in a room with five other people for as little as ₩10,000 or ₩20,000 a night. But do you really want to sleep on a bunk bed in a room with 5 other people every night for 3 months? Do you really want to use a communal toilet and take showers in a communal shower area for 3 months?
And considering that you've been asked to intern for what I can only presume is a decent sized company, they likely wouldn't be located anywhere near the places where one could find cheap hostels. Do you really want to spend 90 minutes taking the subway to your office every morning an another 90 minutes back again in the evening?
Korea isn't Cambodia. In Cambodia, people can get by on a few dollars a day. Meals are cheap. Transportation is cheap. Accommodation is cheap. Everything is cheap. Korea is incredibly expensive.
Yes, you can eat street food, take buses and subways, and find other ways to cut costs. But do you really want to eat samgak kimbap and tteokbokki and Ramyeon every day? Be prepared to drop at least $3,000 over the course of 90 days. Even if you were to live like a monk, there's absolutely no way that you could make it 90 days here on anything less than $1,500, and that's without a roof over your head.
You've got a lot to work out with this.