In 1744, Leonhard Euler published a book with the succinct title Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes. In it, he develops a general method to find curves that satisfy extremal problem, which is cow called the Calculus of Variations. In contrast to the ordinary calculus which allows to find extrema of a single function by solving an equation involving the derivative of a function, here a functional is minimized or maximized over all functions by solving an ordinary differential equation.
His example VII has the title Invenire curvam, qua, inter omnes alias ejusdem longitudinis, circa axem AZ rotata, producat solidum superficies fit vel maxima vel minima.
Euler’s Latin almost doesn’t require a translation into English: To find a curve, which among all others with the same length (meaning defined over the same interval) and rotated about the z-axis, produces a solid whose surface shall be maximal or minimal.
Euler then proceeds, in a few lines, to apply his method to derive the differential equation for finding curves so that the corresponding surface of revolution has extremal area. Euler notes that this equation is solved by the catenary.
I am not a historian, so I do not know who coined the term catenoid, nor do I know who made a first image.
Euler is not concerned with two catenaries passing through the same points and thus offering two different solutions of evidently different area.
Euler neither discusses nor defines the term minimal surface. This is done 1760 by Joseph Lagrange, who establishes in his note Essai d’une nouvelle methode pour determiner les maxima et les minima des formules intégrales indéfinies the minimal surface equation for a graph, observes that planar graphs satisfy his equation, and adds that “la solution générale doit être telle, que le périmètre de la surface puisse être détermine a volonté” –the general solution ought to be such that the perimeter of the surface can be prescribed arbitrarily. Lagrange gives no further examples, but his comment has triggered research that is still ongoing.