I'm writing this partially not to let this question go unanswered and partially to include some details that I didn't find at the MO post.
There is a book by Herman Goldstine titled "A History of the Calculus of Variations from the 17th through the 19th Century" that covers the history and development of the Euler-Lagrange equation (and much more, naturally), and it covers this topic well. In it, Goldstine writes that Euler first discovered what we now call the Euler-Lagrange equation prior to April 15, 1743, which we know as a result of a letter from that date sent by Euler to Daniel Bernoulli containing his discovery. Euler then published this finding to a broader audience in his 1744 Methodus Inveniendi.
Euler's derivation approximated a curve by $N$ points and then let $N$ go to infinity to find extremals. This method was somewhat tedious in its implementation and Euler himself was interested in finding a method that did not rely on any geometry as his method did.
Eleven years later, in a letter dated August 12, 1755, Lagrange (at just 19 years old) sent Euler a letter in which he re-derived Euler's result using purely analytical methods. Lagrange's derivation was powerful enough to handle other types of problems and had Euler's earlier result as a nearly automatic consequence. Euler himself much preferred Lagrange's derivation and gave it the name "calculus of variations," and it is essentially Lagrange's technique that is used today.
The name of the equation, then, is very reasonable given that Euler found it first and Lagrange refined his approach.