Others have brought up possibly redoing the interview, though that carries risks too (two flubbed interviews will make everyone uncomfortable). In this answer I'm going to assume that you didn't hire the person despite the flubs or offer another interview. (I'll come back to "during the interview" at the end.)
That it's an internship is important. The candidate will be looking for another internship (or permanent position) again at the end of this academic cycle. You want to make sure that all your rejected-but-possibly-promising-anyway intern candidates know to re-apply next time. Internships are cyclical; use that.
My company hires a lot of summer interns (about 20 this summer for a team of <300). We're aware that perceptions of our company are important and students talk to each other, so we want people we don't hire to still have a good impression of us. We also want students to remain aware of us beyond the job fairs. Our senior technical people sometimes give talks at local universities; we participate in (and sometimes sponsor) community hackathon events; we are part of relevant users' groups. So for your internship programs in general, look for ways to stay on students' radar in a positive way.
For specific candidates, a couple points:
When you send the rejection, explicitly invite this person to re-apply next {semester, summer, whatever}. This has to not sound like "we'll keep your resume on file" boilerplate; say something positive that makes the candidate think you're talking to her -- something about a particular skill that impressed you, for example.
As the next hiring window approaches, send another message with some information (we'll be at your school's job fair on $date, we're starting to hire summer interns to work on X and Y, etc) and ask her to apply. A little bit of personal outreach can help you stand out from all the other companies the student is looking at.
All that said, you also wanted to know what you could do as an interviewer, presumably during the interview. This depends a lot on how confident you are in reading people (will I make things better or worse?), but I have occasionally had a super-nervous junior candidate in a one-on-one interview, and I've had success with saying something like this: "Hey, it looks like you have some interview jitters. It happens; we were all new once. How about if I give you a couple minutes alone and then we can continue? Would you like something to drink?" This last part gives you an excuse to go somewhere; you don't want to just be lurking outside the door.
Finally, I hope that in all your interviews, intern and otherwise, you're taking some time to sell the company. There's not much that can beat employees who seem to be genuinely happy about their jobs and what they're working on. I've been on the receiving end of this, where I didn't get the job but came away ready to apply again in the future.