What impetus would Einstein have had to believe the speed of light, of all things, should remain constant for all reference frames? If I was in the 1900's, and had to choose between

*

*something as "obvious" as time and space behaving "normally"


*The speed of light staying constant in all reference frames
I would never even dream of assuming the second over the first. What possible impetus could Einstein have had to assume (2.) over something as clearly obvious as (1.)? Was there some huge unexplained phenomenon in physics at the time that would bring anyone to even think about considering (2.) over (1.)?
 A: 
What possible impetus could Einstein have had to assume (2.)
over something as clearly obvious as (1.)

There was Maxwell's theory of electromagnetism.
It also claims that light is an electromagnetic wave,
based on the fact that the measured speed of light ($300000$ km/s)
is equal to predicted speed of electromagnetic waves
($\frac{1}{\sqrt{\epsilon_0\mu_0}}$).
It predicts electromagnetic waves (and hence light) to propagate
with a speed independent of the motion of source and observer.
This clearly contradicts with the concept of absolute space and time,
and the Galilean way of adding of velocities, even though that seems
so "obvious" and natural to humans.

Was there some huge unexplained phenomenon in physics at the time
that would bring anyone to even think about considering (2.) over (1.)?

Yes, there was the Michelson-Morley experiment. It tried to
detect the speed of light to depend on the velocity of the observer.
For this it used the velocity of the earth through space (known to be $30$
km/s) whose direction varies in the course of the year.
But it found nothing. The speed of light did not depend on direction.
