留学: Why the use of "留"? I'm sure like many students of Chinese, my first encounter with the character 留 was in the word 留学, meaning "to study abroad".

Some time later I encountered 留 as an independent word, meaning "to stay". Most of the compounds with 留 are consistent with that meaning (停留, 保留, etc.). However, 留学 doesn't really make a lot of sense as "stay" + "study". The dictionary was not very helpful, as it listed 留学 as a (presumably obvious) example of 留 meaning "to stay in a place".

Could someone proffer an explanation as to how "staying a place studying" came to mean "study abroad" as opposed to, say, "going to boarding school"? Perhaps someone knows something the history of either Chinese education and/or the usage of the word 留学?
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.