The quickest solution is: set environment variable RANDFILE to path where the 'random state' file can be written (of course check the file access permissions), eg. in your command prompt:
set RANDFILE=C:\MyDir\.rnd
openssl genrsa -out my-prvkey.pem 1024
More explanations: OpenSSL on Windows tries to save the 'random state' file in the following order:
I'm pretty sure that in your case it ends up trying to save it in C:\.rnd (and it fails because lack of sufficient access rights). Unfortunately OpenSSL does not print the path that is actually tries to use in any error messages.