Hello Mike,
I looked on www.rootsireland.ie for James Murphy born around 1870 +- 5 years in Co Mayo. This brought up 62 records. As you will appreciate the name Murphy is quite common here so we would need a little more information in order to pinpoint the correct family. We really would need a parents name.
As the timeframe is recent there should be records to access this information. I would suggest that you apply to birth marriage and death records office for New York and get his marriage certificate. This will yield much information. You may need to go back step by step through your parent's and grandparents certificates in order to get the correct information.
We would love to be able to help you trace your ancestor over here but we need a certain level of information. If we could have his parent's names it would make things simple. Have you looked in the local library in the district where he died to see if there is mention of him in an obituary? It's amazing the amount of information contained in these.
Please get back to us when you have a bit more information that we can go on.
Look forward to hearing from you soon,
Kind regards,
Anne Dennehy