The first question we ask students who are interested in doing a marketing or advertising internship is this:
"Here is a pencil. Tell me ten things you can do with this pencil besides writing"
It's one of those things that never fails to sort out ambitious accountants from true marketers. I did this recently and had the student get angry: 'You can't use pencil for chopsticks!'
I had another student today who answered the question brilliantly. He suggested using it as a skewer, a tac, chopsticks (!), a hair pin etc... etc... He's someone we can work with. We accepted him for a paid internship on the spot, and turned down the other guy for an unpaid internship.
It's not like this type of question doesn't come up. In my experience, it's often the deciding question at an interview. Maybe it's not about pencils, but it is about creativity. Host companies will ask 'what are three innovative ways for you to market our product?'
If you're looking for an internship with a marketing company, you need to answer this question in your application.
Before all that of course, you need to find marketing companies. If you look on Craigslist, you're usually wasting your time. You're in a pool of hundreds of other applicants, and you'll be drowned out. Best is to find fresh companies and pitch the idea of an internship to them. Doing this, you not only face little competition, but you get to make your own rules, and structure the internship to more closely match your needs and ambitions. So to find companies, the easiest thing to do is to try Google Maps. Obviously, first try those few companies that you know and are interested in - but when you're done with that, then try Google Maps. It's REALLY effective. Go there and type in something like ' marketing + [insert your town here]' and you'll get a list of dozens and dozens of marketing companies in your city.
Start with that list and get in touch with them. Start with a phone call. People are incredibly receptive to young, ambitious students. People want to give back, to help out - you should structure your pitch that way. Don't give in to the temptation to sell yourself as cheap labour, do that, and well, you'll end up with the kind of person who's attracted to that sort of thing - i.e. a poor, shady and cynical bum.
Once you make contact with a company, be prepared for rejection. Most people will say 'no'. But some will say 'yes', maybe one in ten or one in twenty. Not 'yes' to taking you as an intern, but 'yes' to taking a look at your resume. So you'll get an email and a name, and an invitation to submit your resume.
When you have this contact information, use it right away. If you delay it 2 or 3 days, you won't get the placement. They'll have easily forgotten about you well before then.
This email, presenting yourself, this is key. This is where you need to first show off your creativity.
When we talk to companies on behalf of students, we make a mini pitch to them. We say that we've asked the student to review the site, and they've come up with a few ideas, and then we go over 1 or 2 of them. I.e. 'Student X saw your advertising campaign for that brand of chocolate, and she said that she'd love to join in on something like that. Her idea was to work on branding the actual chocolates themselves with a marzipan star, because she felt ... ' ok, maybe that's not the best idea, but it's enough. It's enough to show you have a living brain, and the courage to use it, and that's enough to get you in the door for the interview.
Beyond that, it's up to you.