Is your cousin new to the city? Are you far from your wife's workplace? Is the cousin also an introvert, or quite social?
I could see a few possibilities here:
1) Cousin may spend most evenings of the summer being active / exploring a new city: summer sports leagues, running / biking / hiking, spending time with other interns. So, he may be less of an imposition than you think, simply because he isn't there a lot.
2) With legitimate reasons not to fully engage with him (house isn't really set up for another person / wife's schedule is not consistent / differs from yours / etc) you could help him around the edges: Stay with us for 2/3/4 weeks until you find another intern to room with, or sublet in town close to work / what you want to do; or have your wife contact the intern coordinator with HR, and see what the interns typically do. If he's lucky, maybe there are corporate apartments available for those out-of-town, or maybe they have a list of interns he could contact to look for roomies.
3) If there is nothing through work, is there another situation that you see you could make a match or otherwise help? Local college with lots of summer sublets available cheaply / church or other social unit where you could ask if someone is looking to let out a room, or house sit for a season, etc.
You can be very helpful without going all out. There is nothing wrong with having limits. Have him over for dinner a few times, though. Just helping him get settled will help him make his internship a success, which is the whole point.
If he is really having a hardship paying for summer accommodations, you could consider more. That really shouldn't be an issue with an engineering internship, which should be paid fairly well, particularly these days. But if he is paying for college on his own, he might also be looking to save up money for the rest of the year, too.