What are some prestigious research internships that look good on college applications?
What’s your major? And I’m assuming you mean for grad school? REUs always look good, because they’re pretty selective and overseen by the National Science Foundation.
Are you in high school? If so, I think ANY research is helpful! Let me know if you want any tips on reaching out to local universities for research opportunities!
If you could give me some, that would be awesome! 
Research anywhere at a college nearby is great as long as you show you did something legit.
Also i think some big name organizations like NIH and FDA have research internships; One of my friends interned at NIH her junior year summer.