I realize you can’t budget for it - but let me just mention that some Universities might offer modest fellowship grants, and not all students will put the effort into yet another application/essay.
My daughter was a first-year, and never expected to have a “shot” at it - yet she was awarded enough to cover the air travel.
We considered this, and I looked at the study abroad programs at a few schools that we were considering. It seemed like pretty much every university that we looked at has some sort of study abroad program. Details seemed to vary. Quite a few of these were reasonably priced.
I agree with other comments to just pick some universities that you are already considering, and start with their web pages and see what they offer.
One daughter studied abroad for one semester in high school. All classes were in Spanish, except for the school’s English class where my daughter became almost an assistant teacher. The only cost to us was the cost of the air fare, which we did need to pay for. Our daughter stayed at the home of a girl who was the same age. Then both of them came to our house and the girl from South America lived with us for approximately a semester. The whole thing was very, very positive experience.
The same daughter studied “slightly abroad” for the full four years of her bachelor’s degree (we live in the northeast of the US, she got her bachelor’s in Canada). She got a great education at a very good university, got the experience of living abroad for four years, and we saved quite a bit (it was less expensive than our in-state public school would have been even with a merit scholarship). I wondered if she would want to “study abroad” back near here for a semester, but she got involved in research on campus that she wanted to continue to work on.
There can of course be reasons that could come up part way through why a student might not want to study abroad. Getting involved in research or internships locally might be one reason.
I agree with other answers that suggest considering short courses over the summer. Our daughter took a couple of short courses over the summer. One was a 5 week intensive French immersion program at a different university. She did get a scholarship which covered most of the cost, including room and board. The other was a two week offsite biology field study.
I agree with @gardenstategal that there may be some cases where some of these programs cost less than just attending your home university.