I honestly have no idea, but sort of curious now too xD
The move c4 is played, so maybe it references the English Opening? Perhaps "Rat" was the last name of the GM who popularized the opening, or something?
Either this, or maybe the tricky guy who invented this for the black side is from England and his opponent called him a "Rat" for tricking them into 3. dxe5 dxe5 4. Qxd8+ Kxd8 as a rare case where black is slightly better, despite being the one to lose castling rights 
Eric Schiller. "The Rat" is a Canadian name for the "The Modern" : 1.e4 g6 2.d4 Bg7. Did anybody in England also call 1.d4 d6 2.c4 e5 "The Rat"? That would explain why it's the "English Rat", but actually Schiller gave names to many openings without any reason whatsoever. I wouldn't think too hard about it.
Eric Schiller. "The Rat" is a Canadian name for the "The Modern" : 1.e4 g6 2.d4 Bg7. Did anybody in England also call 1.d4 d6 2.c4 e5 "The Rat"? That would explain why it's the "English Rat", but actually Schiller gave names to many openings without any reason whatsoever. I wouldn't think too hard about it.
That's interesting
Eric Schiller. "The Rat" is a Canadian name for the "The Modern" : 1.e4 g6 2.d4 Bg7. Did anybody in England also call 1.d4 d6 2.c4 e5 "The Rat"? That would explain why it's the "English Rat", but actually Schiller gave names to many openings without any reason whatsoever. I wouldn't think too hard about it.
Eric Schiller....... what a character! The guy wrote more chess books than there are possible moves in chess!
R.I.P. Eric.
He wrote many books but the quality of the books was extremely low.
My own theory is that a rat jumped onto a chess board during a game in Englund around that certain position of the opening, scaring the heck out of everyone there except the player who got the idea to name the opening after the rat out of amusement.
How bad were the books?
My own theory is that a rat jumped onto a chess board during a game in Englund around that certain position of the opening, scaring the heck out of everyone there except the player who got the idea to name the opening after the rat out of amusement.
Indeed, that is also the origin of the word 'mouseslip,' which occurs when you blunder due to being startled by a rodent.
If anyone was wondering, this is the English Rat Defense:
What I don't know is how did anyone ever came up with this name. Feel free to share what you know or any theories if the origin is still a mystery.