The following from Francis H. Gardener's Early Schools and Teachers of Lyons (1909) - 

"Authorities differ as to the location of the first school house in the Village of Lyons. One says that the very first school building in the place, and probably in the town, was a primitive log structure that stood on the hill on the west side of Butternut Street at the head of Queen St. It was there, we are told, as early as 1804 or 1805, but was burned down soon afterward. In June 1813 the town was divided into twelve school districts. J.W. Gillespie and John Brown were school commissioners. It seems impossible to find the names of any teachers who taught in this ancient school edifice or of the pupils attending school there.

Grip's Historical Souvenir of Lyons says that the first school house in the village stood on the west side of Butternut St. just south of present jail on the site of the old Catholic Church. It was built of logs at first and Thomas Rogers taught there in 1808. This was afterward replaced by a frame building, the fifth one erected in the Village for school purpose."

Note - this paper contains a good bit of additional information about schools in the Town of Lyons.