Since John Browne had been the Mayor of Houston, Texas we first searched The Political Graveyard and discovered that John T. Browne had been Mayor of Houston from 1892 to 1896. According to The Political Graveyard he was born in Ireland on March 23, 1845, married Mollie Bergin on September 13, 1871 and died in Texas on August 19, 1941.
Next we searched the Public Member Trees at Ancestry.com and learned that John's wife Mollie was named Mary Jane (Molly) and that she was born in New Orleans, Louisiana in 1850 and died in Houston, Texas in 1935. Her parents were Michael and Mary Jane Crabb Bergin.
The couple had the following children: John, Thomas, Joseph, Mary, Francis, Winifred, Jane, Margaret, James, George, Johanna and Catherine. These children were born between 1872 and 1891
We next searched the Historical Newspapers, Birth, Marriage and Death Announcements (1851-2003) database at Ancestry.com but found nothing. So we looked at the 1880 US Census where we found that the family was living in the 5th Ward of Houston, Harris County, Texas. John's occupation was listed as grocer. In the 1900 US Census John's occupation was listed as Legislature. According to the 1910 US Census the family was living on Lynwood Ave in Houston.
Finally we went back to The Political Graveyard and saw that John Browne was nicknamed "The Fighting Irishman" and "Honest John" and that he was buried at the Glenwood Cemetery in Houston, Texas.
In many cases you can find the birth date of a deceased individual on his or her tombstone at the cemetery where he or she was buried.
You can go directly to the cemetery and transcribe the tombstone information or you can shoot a photo of the tombstone.
You can also try to find the tombstone information online as there are some sites that concentrate on compiling this information. Here is the link to The USGenWeb Tombstone Transcription Project.