You asked for a birth record, which I did not locate. Working from the helpful details you and/or others have posted to profile Small-1934, I both (a) was not able to confirm the birth of Ethel Small aboard the Northampton in 1879 and (b) reviewed some information that only might contradict such a claim.
Commenting here with I hope helpful information. You may be able to make a more careful review of the record I examined and learn more than I did.
The Northampton's arriving passenger list, dated 12 January 1880, has been indexed and digitized. These results appear on the site, Mariners and ships in Austrailian Waters. Entry for the indexed and digitized materials cites "State Records Authority of New South Wales: Shipping Master's Office; Passengers Arriving 1855 - 1922; NRS13278, [X146-147] reel 441. Transcribed by Cheryl A. Lean."
Transcription with links to the digital images appears as "Northampton: of London, John C. Clare, Master ..... from Port of London to Sydney, New South Wales ..."
1. Two men, Thomas Small, appear on the arriving passenger list. One is reported age 41, of England, a cooper; has apparent wife Susan. They have several apparent children. The other Thomas is age 16, of England; he is a carpenter whose entry is listed with an "Alfred Small, age 14, "chemist asst." Also arriving and separately listed is an Ada Small, age 12, of England; "servant"
2. Small-1934 is associated on WikiTree with parents Thomas Small and Susan _____. The arriving passenger entry about these fine folks appears as scanned page 6 (of 13). No such arriving passenger named Ethel Small or infant Small is reported on the list.
Of note, in my review of the scans, I did not find a reconciliation of departing and arriving passengers that might include a list of those who died or were born on board.
All sources are subject to error, omission, etc. If your careful review of these items does not uncover more information about Ethel's birth, consider approaching the problem from two angles, which you may be doing already:
First, in the chance she was born aboard the ship, seek to learn any other notation about the passage that might exist. This would include what are probably the English departure records. You could query the Australian archives to learn if there are other sheets or papers associated with this arrival, especially those that might include a reconciliation to the departure list departures. You might inquire of the protocol for the arriving lists. (See also Peter Knowles helpful comment.) As well, it would include trying to learn the earliest notiation about Ethel that reported she was born aboard the Northampton.
Second, take the approarch that something is amiss with the notion that she was born aboard the ship. This would include seeking records over time about her age and birth specifics as well as other work that would disprove or further conflict the notion of birth onboard.
I am not as experienced with Australian records as others surely are. If this were a case in the U.S., I would research to learn the information about her found recorded in her children's birth, marriage and death records.
Hope this helps. --GeneJ
 Many births and deaths occurred aboard ships, eonough so to expect commerce and migration in 1879-1880 had procedures for the reporting. These might have been part of the port procedures (both departure/arrival).