Before the fiction came the science. Cathode rays were first discovered in 1867, and named in 1876; in 1895 Wilhelm Röntgen discovered the "x-ray", which caused an explosion of interest in radiation, science and pseudoscience. The tremendous number of articles and essays about the discovery of these rays - particularly x-rays - and the exceptional properties ascribed to their use set the stage for what we would call early science fiction, though it was often called "scientifiction" or "pseudo-scientific stories." So even before the appearance of ray guns, rays were a part of speculative fiction, because they were a part of the popular understanding of physics. So in Ambrose Bierce' story "The Damned Thing" (1893), you have passages like:

At each end of the solar spectrum the chemist can detect the presence of what are known as "actinic" rays. They represent colours--integral colours in the composition of light--which we are unable to discern. The human eye is an imperfect instrument; its range is but a few octaves of the real "chromatic scale." I am not mad; there are colours that we cannot see.

The first reported "ray" weapon in science fiction was the "heat ray" used by the Martians in The War of the Worlds (1897):

Then suddenly the white flashes of the Heat-Ray came leaping towards me. The houses caved in as they dissolved at its touch, and darted out flames; the trees changed to fire with a roar. The Ray flickered up and down the towing path, licking off the people who ran this way and that, and came down to the water's edge not fifty yards from where I stood. I t swept across the river to Shepperton, and the water in its track rose in a boiling weal crested with steam.

This is, of course, only the modern ray weapon; if you care to stretch a point you could include the description of Archimedes' solar ship-burning weapon from the siege of Syracuse.

Wells' idea was literally followed up by Garrett P. Serviss in an unofficial "sequel" to Wells' book, Edison's Conquest of Mars (1898), and others followed suit. The raygun is first attested in 1917, in Victor Rousseau's The Message from the Cylinder - which also gives some great descriptions of the thought processes that went into rays back then (92-93):

["]We were on the point of compelling her to come in when she discovered the secret of the Glow Ray, which would have made the effort unremunerative."

"What is this Ray?"

"It is combustion by old light, stored solar energy being transmitted for that and all other power purposes from the great solar works on the Vosges Mountains. Its invention made the old warfare obsolete. Our small-arms are miniature Ray mirrors, charged with a single unit; our big ordnance is supplied from the Vosges by cable connection. The Ray destroys everything that it encounters, not protected by the glow paint which you may have observed n the fronts of our buildings. This is the last and greatest of the cola tar discoveries, and its manufacture is based upon the exact relationship between the disintegrating glow rays and an exact color having a fixed number of vibrations.

The laser, as far as the concept of stimulated emission, was also conceived in 1917 by Albert Einstein, although the actual term wouldn't come into being until 1959.

A lot of this basic material is covered in the Science Fiction and Fantasy Encyclopedia's entries on Ray Guns and Rays, which are decent places to start if you want to explore further. (Which seems dangerously close to citing Wikipedia, but the SF&F Encyclopedia was at least a hardcopy book). If you want more on the physics history leading up to it, I would recommend How the Ray Gun Got Its Zap: Odd Excursions Into Optics.