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I would say any answer must be guesswork, so I will make this a community wiki.
Hash, or hash browns, is breakfast food made from cutting potatoes into long thin strips (smaller than french fries, and shorter, but proportionally similar), then frying the mass of strips in animal or vegetable fat until browned, stuck together, and cooked. By analogy, 'hashing' a number meant turning it into another, usually smaller, number using a method which still deterministically depending on the input number.
I believe the term "hash" was first used in the context of "hash table", which was commonly used in the 1960's on mainframe-type machines. In these cases, usually an integer value with a large range is converted to a "hash table index" which is a small integer. It is important for an efficient hash table that the "hash function" be evenly distributed, or "flat."
I don't have a citation, that is how I have understood the analogy since I heard it in the 80's. Someone must have been there when the term was first applied, though.