Julia Robinson's 1949 paper "On the Hamiltonian Game (A Traveling Salesman Problem)"  begins with the sentence: 
"The purpose of this note is to give a method for solving a problem related to the traveling salesman problem."
Although the problem was apparently well known at that time, there does not appear to be any earlier reference in the literature.  Solution methods began to appear in papers in the mid-1950s; these early papers used a variity of minor variations of the name traveling salesman problem.  Dantzig, Fulkerson, and Johnson (1954) referred to the "traveling-salesman problem",  Heller (1954) used "travelling salesman's problem",  and Morton and Land (1955) preferred "the `travelling salesman' problem" (and write that they orginally called it the "laundry van problem"). 

We follow Robinson and use "traveling salesman problem".  The sixth edition of The Concise Oxford Dictionary offers some support, writing "travelling-bag", "travelling-cap", and "travelling clock" all with two l's, but "traveling salesman" with a single l.  The second edition of The Oxford English Dictionary does make this distinction, however, writing "travelling salesman problem" (despite a reference to Dantzig, Fulkerson, and Johnson (1954)).