The medicine's name was said 'Six-sixty-six', not 'six-six-six'. Radio ads for the medicine had the following jingle, sung to the tune of Farmer in the Dell:

Six-sixty-six
Six-sixty-six
Colds make you feel so bad,
Take Six-sixty-six

Source

It also explains why it’s called that (it was the next number in the original pharmacist’s prescription book).