As her birth reg confirms, Jane Ann Smith’s mother’s maiden name was Goodwin.
By looking at GRO birth registrations with the last name Smith and mmn Goodwin, I was able to identify the birth registrations for the siblings who appear with Jane on English census returns in 1861 and 1871. There is no John Henry / Henry John / Henry / John on the census returns, and no birth registration for any of those names with a last name of Smith and mmn Goodwin in 1866/7/8.
She did have a brother called Harry Frederick, who the profile manager thinks is the Henry in the immigration record (in which case they were probably understating his age for a cheaper passage). But he died in 1948.
I don't know anything about Australian records, but I noticed that Jane Ann's QLD death registration gave the names of her parents, including her mother's maiden name. Might that also be the case for John Henry?