Although the title for this YouTube vid says "1965", this site says that Del Shannon's "Runaway" was the number one song the day I was born, in 1961.
Here is the number one song for my dh's birthday in 1955. It happens to be one of his favorites. "Autumn Leaves"--the instrumental version by Roger Williams. The only YouTube I could find of it has it set to a visual of Disney's Pocahontas.