What do you do if you want to impress game design companies with your internship application?

It's important to consider when building your portfolio that you not only want to showcase the most impressive titles, but also a selection of both mainstream and niche projects to highlight a wider range of use-cases for your own skillset. If you only have a specific set of genres you work in, then highlight that above all else so that the hiring party sees you as though you were made for the role, which would be true. Also, provide a number of titles that you've worked on, as game design relies very heavily on having a broad set of experiences. If you've been on one project, you might lack the foresight of design issues with a certain feature. If you've been on ten, highlight it so that the hiring party doesn't assume the former option.

Specifically mention the roles you worked most closely with, highlighting that you worked with artists to get visual feel nailed down for a certain feature or level, or that you worked with engineers from a technical perspective to get gameplay feel nailed down. If you've done both, highlight both. Be very specific about what you designed and include examples of it where possible so that your work speaks for itself. Design is always a team effort, so be clear about your team experiences and willingness to iterate many many times.

If you're applying for a specific project, be clear about your predictions for who will be playing the project at release and why - including cautions or easy wins. However, most cases will be applying to a company as a whole, so you don't want to make too many bold predictions about projects as it comes off as favoritism (which isn't a good look as a designer, despite bias being everywhere) and lowers your versatility. The important things to mention, if true, are that you're keeping up with new title performances in the industry and staying informed via X or Y on what does or doesn't perform for them.

Personalization helps you stand out for a role as the tailor-made candidate. A plain resume works wonders for large companies, but any team under 100 people will likely have a real human sorting through the resumes, allowing you an opportunity to shine. It can be tedious, but personalized applications really do set you apart from the crowd.

Express your passion strongly and make it clear how you feel about the industry / career, but be wary to not make too much of your application or resume about passion. Passion is what drives us to do well, but "too much passion" can mask both a designer's ability to iterate (becoming married to ideas) and ability to work in a larger team. Passion should be at least 25% of the "feel" of your application, but shouldn't be more than 5-10% of the actual tangible content.