留学: Why the use of "留"?
The answer is in this article:
从"出洋"、"游学"到"留学"——晚清"留学"词源考

It is really a complicated history, but in short: During the Tang Dynasty Japanse students came to China to study. These were called 留学生 in Japan. During the Late Qing Dynasty 留学 was taken over in China and became more and more popular.