Common law marriage cannot be established in New York or Michigan. Unless she had a will that states otherwise, her next of kin inherit her property. If his name is on an apartment lease, car, and joint bank account, then he certainly can claim those items but it is possible her heirs can make a claim as well.
I am a NY lawyer. His chances are not good at all based on the facts as you presented them. The NY courts are very strict about making someone prove if they were arried or not. The greater question is where was she domiciled. A decedent can only be domiciled in one place. A NY court will only take jurisdiction over a NY domiciliary.