The prediction that electrons should exhibit wavelike properties was supported by what evidence?
The evidence comes from the double slit experiments
In the mid 1920's Davisson and Germer were studying how a beam of electrons reflected off of nickel. The nickel crystals acted like a the two slit experiments. A pattern of interference like that of water waves emerged. The experiments of Davisson and Germer showed that electrons exhibit interference phenomena a property of waves.
( The Elegant Universe Brain Greene page 104 1999. )
When water waves go through two slits they interfere with each other creating patterns of dark and light. Electrons going through two slits create the same inference patterns.
The double silt experiments provide evidence that electrons function as waves as well as particles.