Most people meet this word as the name of a library, the collected rabbinic commentary on Scripture. That is a late use. The word itself is much older and much simpler: midrash is the noun from darash, to seek, to inquire, to search out. It is the activity, not the shelf.

The word appears exactly twice in the Tanakh, both times in Chronicles, and both times as a written record: the midrash of the prophet Iddo, 2 Chronicles 13:22, and the midrash of the book of the kings, 2 Chronicles 24:27. Translators render it story, or commentary, or annotations, and none of those are wrong. What they miss is that the thing being named is the product of somebody having searched.

Hold that against how we usually treat study. We go looking for the answer so that the searching can stop. Midrash names the searching itself as the thing worth writing down.