That depends on what you want to do within the video game industry. The good news is almost anything can work.
Broadly, there are 5 career branches in the industry:
Software Engineering: these people write the code that makes everything work. One obvious major is Computer Science, but that's not the only one. I have been a software engineer at several studios and my majors were Physics and Pure Mathematics. I was self-taught at programming. I have known excellent software engineers who dropped out of college and some that didn't even finish high school. They were all excellent coders, though, so to go down this road, you must learn programming somehow.
Game Design: these people build the levels, write the dialog, and develop the mathematical models that define the game. As you might guess, a broad range of majors can apply here. Some people have degrees in Economics, others in English Literature. I have also been a Game Designer at many studios, where my background in Mathematics came in handy. Here again, though, the key factor is a love and understanding of how games work. Most game designers I know are huge fans of all kinds of games: board games, card games, party games, etc -- not just video games.
Visual/Audio Art: Most of the great video game artist I know have a background in classical fine arts. It's not always painting or sculpture: one of best animators I know had a degree in metallurgy. The key here is to understand the fundamentals related to the field in which you want to excel: composition for graphic artists, principles of animation for animators, etc. If you're going to do sound work, a degree in audio engineering is a great place to start.
Production: producers build teams and keep them on track. It might sound boring, but they are the oil that makes teams run smoothly. They must understand how to build schedules, predict timelines, and manage budgets and people. Producers can come from virtually any background, but business classes can help them understand the business side of the process in ways the other team members often don't.
Business Development: studios that self-publish their games have one or more business development teams. These consist of marketing and salespeople. If the publisher is big enough, the business team probably also includes artists and technicians who build and maintain the company website and all business-related infrastructure. Degrees in marketing, communications (for public relations), and IT all have a home here.
So, really, almost any degree can be "game-related" these days. The trick is to network with gaming companies and be willing to move where the gaming business thrives: most of coastal California; Austin, TX, and Seattle, WA. There is a growing presence of studios along the East Coast, too.
This professional recommends the following next steps:
- Pick a major that hits the sweet spot between the highest-quality education and what you ultimately want to do.
- Figure out which of the 5 branches of game dev you find most interesting.