ネーム is the rough draft which the finished manga is based on. (reference)
I can't for the life of me figure out the origin of this word. Why did it end up being the English word "name"?
So you're referring to ネーム in this sense, right?
ネーム was originally jargon used in the Japanese (non-manga) publishing industry, where it means 'caption' (of figures, tables, etc.), a short text describing the essential point of a figure/table. Then it gained a broader sense, and it may also refer to any "floating" small fragment of text that has to be typeset separately from the main text — such as contents of balloons of manga.
The following is my speculation, but in the past when DTP was not available, manga publishers needed the contents of the balloons before the drawings, because they needed to physically prepare the movable types. People in charge of the typesetting process might have said "絵は要らない! 早くネームをくれ!" ("I don't want pictures! Just give me the ネーム!").
When this term, ネーム, came into use in the manga industry, its meaning changed a bit. It began to refer to not only the balloon contents, but also the storyboarding/plotting process as a whole. Even today, the most important part of ネーム is the text, not pictures, as you can see in real examples.