Name, year of registration, country of birth, from search on New Zealand BDM website in June 2025:
Reg. 1880/347. Jones, William Henry. Mother: Hannah. Father: Thomas

Full date of birth of 9 December 1879 from further searching in June 2025 on BDM Search by incrementally changing the date range for the birth.

Place of birth of "Auckland" from the New Zealand Birth Index found in June 2025, so could be ours:
Name: William Henry Jones
Registration Quarter: Jan-Feb-Mar 1880 [Registration]
Registration Place: Auckland, New Zealand
Folio Number: 48

Was to be verified this is a child of our Thomas JONES and Hannah SAMMELS.

UPDATE: Is the son of Thomas JONES and Hannah SAMMELS as per William’s Birth Certificate received in June 2025:
Birth District: Auckland
Date Born: 9 December 1879
Where Born: Nelson Street, Auckland
Name of Child: William Henry
Child With Parent at Time of Registration: Not
Sex: M
Father's Name: Thomas Jones
Father's Rank or Profession: Sawyer
Father's Age: 30 [Was in fact 40]
Father's Birth-place: Cornwall
When and Where Married: 1867, Kenwyn, Cornwall
Mother's Name: Hannah Jones
Mother's Maiden Name: Sammels
Mother's Age: 30 [Was in fact 33]
Mother's Birth-place: Cornwall
Informant Signature: Hannah Jones (her mark +)
Informant Description: Mother
Informant Residence: Auckland
Name, if added or altered after Registration of Birth: [Blank]

New Zealand Electoral Roll - Auckland Central Roll for 1919, found in June 2025:
Jones, John, 1 Canada Street, labourer [Subsequent Rolls has this correct as no. 2 Canada Street]
Jones, Grace 2 Canada street, married
Jones, William Henry, 2 Canada street, railway worker [Is this a brother of John Jones in this entry]?