I looked up James the son in the 1910 United States Federal Census and according to this both parents were born in Ireland.

Name: James Tinkler
Age in 1910: 4
Birth Year: abt 1906
Birthplace: New York
Home in 1910: Brooklyn Ward 28, Kings, New York
Race: White
Gender: Male
Relation to Head of House: Son
Marital Status: Single
Father's Name: James Tinkler
Father's Birthplace: Ireland
Mother's Name: Annie Tinkler
Mother's Birthplace: Ireland

James Tinkler 38
Annie Tinkler 37
George Tinkler 12
James Tinkler 4

States on the above census they had been married 16 yrs in 1910 had had 5 children only two still living. Both entered the USA in 1885.

New York, Naturalization Petitions, 1794-1906
Name: James Tinkler
Birth Date: abt 1872
Birth Place: Ireland
Age: 30
Record Date: 20 Nov 1902
Court: County Court of Kings County (1-333)
Court Location: New York, Kings, New York, USA
Title/Description: County Court of Kings County (320-321)
ARC Number: 5324244
ARC Title: Petitions for Naturalization, 1793-1906
Record Group Title: Records of the Immigration and Naturalization Service
Record Group Number: 85

At this point james was still single and his occ. clerk.

1900 United States Federal Census
Name: James Tinkler
Age: 29
Birth Date: May 1871
Birthplace: Ireland
Home in 1900: Brooklyn Ward 3, Kings, New York
Race: White
Gender: Male
Immigration Year: 1888
Relation to Head of House: Head
Marital Status: Married
Spouse's Name: Annie Tinkler
Marriage Year: 1894
Years Married: 6
Father's Birthplace: Ireland
Mother's Birthplace: Ireland

James Tinkler 29
Annie Tinkler 27
John Tinkler 4 **
Geo Tinkler 1
Sarah Tinkler 4 **

** must be the children who died.

Married 6 yrs arrived US 1888

U.S., World War I Draft Registration Cards, 1917-1918
Name: George John Tinkler
County: Windsor
State: Vermont
Birth Date: 23 Nov 1898
Race: White

Give Occ. Miner next of kin James Tinkler - Brooklyn, New York

U.S. Veterans Gravesites, ca.1775-2006
Name: James Patrick Jr Tinkler
Service Info.: S/1C US NAVY
Birth Date: 6 Oct 1905
Death Date: 21 Mar 1959
Service Start Date: 15 Dec 1923
Interment Date: 24 Mar 1959
Cemetery: Long Island National Cemetery
Cemetery Address: 2040 Wellwood Avenue Farmingdale, NY 11735-1211
Buried At: Section T Site 1839

I think this is the son of James (Patrick ?) Tinkler and Annie ???????

I think was his wife:

U.S. Veterans Gravesites, ca.1775-2006 about Catherine Tinkler
Name: Catherine Tinkler
Service Info.: S/1C US NAVY
Birth Date: 25 Aug 1905
Death Date: 23 Jan 1979
Relation: Wife Of Tinkler, James Patrick
Interment Date: 25 Jan 1979
Cemetery: Long Island National Cemetery
Cemetery Address: 2040 Wellwood Avenue Farmingdale, NY 11735-1211
Buried At: Section T Site 1839

Do you have any idea of Annie's maiden name and where she and James married. Could it be Hines ? Do you know if James middle name was Patrick.

Thank you, noggin, for your detailed reply.
I do have some of the records you listed..and some are contradictory....as usual.
The closest records are the 1900, 1910 census. The Naturalization request seems to be a second SINGLE James.
As for Annie's maiden name: the real one I'm looking for is Mary Ann Kavanagh and
I believe they married in Ireland ~1893/4-they left a baby behind with g.parents.
The two sons are my strongest clues.....but I have no idea where to from there.
But I'll plod on with them.
Thanks again.

My Grandmother Sarah Tinkler Thomas was the daughter of James Tinkler and Annie Kavanagh. Her two brothers were George and James. She stayed behind in Ireland until she was almost 17 years old. She lived with her Grandmother Eliza and Uncle Daniel in Portarlington. She always told us that she never found her family when she got to America.