As we know, the revolutionizing step of the Special Relativity was, to solve the contradiction between electromagnetism and the Newtonian mechanics, not the Electromagnetism should be altered (which, as a new thing, had been the "natural" option), but the Newtonian mechanics.
But the problems leading to that existed long before (f.e., but not exclusively, the Michelson-Morley experiment).
Does any concurrent theory existed at the time, which was also based on this surprising idea?
Did any of them had signs, with time it could have developed into a Special Relativity, if Einstein didn't exist?
P.s. Aether and aether-dragging theories are not okay - they tried to solve the contradictions not by changing the classical mechanics. They suspected a new form of matter, which has uncommon properties, but behaved on the classical mechanics.