I've just discovered a few refs to an American (?) humorist called Harry Graham, who published a book of the above title in 1899. I must see if I can track down a copy. Here are a few examples why:
"Billy, in one of his nice new sashes,
Fell in the fire and was burnt to ashes;
Now, although the room grows chilly,
I haven't the heart to poke poor Billy."
"Little Willie shot his sister,
She was dead before we missed her.
Willie's always up to tricks.
Ain't he cute? He's only six."
"Willie saw some dynamite,
Couldn't understand it quite;
Curiosity never pays:
It rained Willie seven days."
1899! Makes Hilaire Belloc read like a Ladybird book. According to a note I've just found on the net, it was Graham's Little Willie rhymes that inspired the phrase 'to give one the willies'. Of course, that has more than one usage now... ahem. Incidentally, if I wanted to buy the first edition (illustrated above) I found on sale online, it would cost me £375.