Middle name is William per Family Search but doesn't appear to be transcribed on any official records.
FamilySearch also shows a daughter, Florence E Honary b. 1867 who married Edward Elliott in 1897. There are no associated birth records which could apply. There is a relative marriage record (NSW BDM reg. no. 1146/1897, of ELLIOTT, EDWARD R, and HONNERY, FLORENCE E, registered at GUNNEDAH). There are also no apparently relevant death records in either NSW or Queensland between 1897 and 1988. On balance it seems hard to include her here at this point.
Find A Grave, database and images (https://www.findagrave.com : accessed 24 March 2019), memorial page for Henry William Honnery (unknown–28 Mar 1915), Find A Grave Memorial no. 171548025, citing Chinchilla Pioneer Cemetery, Chinchilla, Western Downs Region, Queensland, Australia ; Maintained by Carol M (contributor 48502914). Find A Grave: Memorial #171548025
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DNA Connections
It may be possible to confirm family relationships with Henry by comparing test results with other carriers of his Y-chromosome or his mother's mitochondrial DNA.
However, there are no known yDNA or mtDNA test-takers in his direct paternal or maternal line.
It is likely that these autosomal DNA test-takers will share some percentage of DNA with Henry: