I read an article earlier today that described a young man at Stanford University as “a rising junior.” What does “rising” mean in this context? The article with the wording is here.
I’m thinking “rising” as in academic achievement and prestige.
“Rising” means a student in between one year and another, with the year that they are rising to what they will be when school starts up again. “Rising junior” means that the person in question has completed their sohpomore year of schooling and will be a junior in the fall.
Sports Illustrated uses this term constantly to describe student athletes in their summer issues. They use it more often for high schoolers than college students, though.
At the high school in which I taught, “rising” was used mainly to describe the incoming freshmen, that is, 8th graders who were almost done with their middle-school lives and, come the fall, would be enrolled as freshmen. We never used it to describe any other group of students, such as those who would start the fall as sophomores; once you had completed the credit requirements for a “sophomore”, you were called that, regardless of whether the new year had started. But some people see it as a cute way of talking about someone during the summer, I guess.
Same descriptor today in stories about the Smith College student who was approached by police while eating in one of the buildings.
The term was very common in my kids’ high school when talking about things that would happen the following academic year.
They use it all the time here, referring to students who are still in one class but will be moving to another class in the fall. I usually start seeing it around April, when housing assignments begin. A rising sophomore would be currently a freshman.
It would be inconsistent to change only one name, but that particular example could be covered by “sophomoreorless”. 
What others have said. It’s somebody who was a sophomore and will be a junior when school resumes.