In addition to the new program that @CT1417 listed, Yale also offers an International Summer Abroad subsidized internship in addition to a broad range of fellowship and grant programs to fund summer internships.
Connecticut College does this. All you have to do is work with the career office to set it up (to ensure that it worthwile for the student.) I recall it was $3600 for the summer. Anyone could do it, and it could be for any kind of work - not necessarily not for profit, etc. Its availability to every student made it a standout!
Wesleyan’s is similar to Yale’s, but begins sophomore year: http://www.wesleyan.edu/careercenter/students/wesleyan-grants/summerexperiencegrant.html
@OHMomof2 - Yes, you had to pay to get credit for the (required) internship. The fee for credit for a summer internshp was over $3000 and the student still had to pay for transportation and living expenses. I talked to the woman in the office that sets up the internships and they were very successful in placing everyone in internships. But almost none of them were paid internships. I left with a feeling that the school was in cahoots with certain businesses to provide free labor.
We much preferred the schools with programs that provided funds to cover living expenses during unpaid internships. 
Skidmore does this as well. I believe they have both a list for local (Albany and Syracuse areas) and allow you to apply for funds for unpaid internships found on your own.
Ugh @My3Kiddos that sounds really onerous.
For sure!
Bucknell has this as welll. But I think only for non profit organizations. There is also a creativity grant program for artistic endeavors.
When my son was at Bates they gave him $1500 (iirc) so he could do an unpaid summer internship for a Judge. He also had a part time job to earn extra money.
The University of Richmond has a program where every undergraduate student is eligible to receive a fellowship of up to $4,000 for a summer internship or faculty-mentored research project.
https://www.richmond.edu/guarantee/
When I was an undergraduate, I participated in a college-sponsored cooperative education (co-op) program, which also provides valuable work experience and college credit while working at an actual employer.
We just visited Ohio State. They said they offered everyone $2,000 after their freshman year for educational purposes. This could be used to study abroad, take an unpiadinternship, work on a research project.
Our tour guide used his to study abroad. Another student described how they wanted to write a book but couldn’t afford to take the time off in the summer as they needed to earn college money. They took the $2,000, didn’t work one summer and wrote the book instead. Sounds like a range of what you can use the money for as long as it is educational. My daughter wants to study abroad and thought this sounded great and a nice start to raising the funds needed.
Not all of these opportunities are advertised. My daughter was turned down for an on campus internship after her sophomore year ( these typically go to juniors), but one of her professors asked her to come up with an idea of her own. This is how she ended up being paid to study the marginalia of an illuminated manuscript at the University of Leeds. She lived in graduate housing, and learned how to cook and shop on a budget. The latter was the most valuable thing to me, as her mother!