Timothy Harrington died October 4, 1896 in syracuse
NYS Vital Record Death Certificate # 41515

Many death certificates contain information such as father's name, mother's maiden name, etc. but not all have that information, it depends on if there is a family member who acts as an "informant". You can get a copy of the certificate from the state, or call syracuse and see if their city clerk has a copy to send you.
If you go the NYS route, let me know - the State will still charge you $22, but I can make the process happen much more quickly.