Although perhaps not originally named after Robert Burns (1759-1796), the poet, balladeer and Scotland's favourite son, this cocktail is drunk to honour...
I found another earlier reference to a Bobby Burns cocktail in an newspaper article titled "Try these new drinks" from the September 9th 1899 issue of The National Police Gazette (New York).
"Bobby Burns Cocktail (by W. J. Burke, Litchfield, Conn). Use mixing glass; fill up with shaved ice; two dashes of gum; two dashes of oranges bitters; two dashes of ginger cordial; one pony of Scotch Whiskey. Strain into cocktail glass; squeeze lemon rind and serve."