RootsChat.Com

I found this:

https://www.greatwarforum.org/topic/183279-liebert-gifford-british-navy/

but when I got to the last post, I realized it was you. To quote: " (Around 1998 or so in the earliest incarnation of the internet, I came across a reference to Liebert Gifford working on a boat sailing from Ireland to, I think it was South America at the age of 16; unfortunately the internet ate the reference later and I haven't found it again."

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I didn't find the voyage to South America.

arrived Canada Quebec, on the Emperor of Ireland, March 1907, but boat and date scrached out and replaced with 1900

the manifest for this arrival doesn't mention 1907 or the ship, but just says he arrived Canada 1900

 Name   Ernest Hansen
Sex   Male
Age   74
Birth Year (Estimated)   1926
Residence Date   11 Sep 2000
Address   5411 Greenough Way
Death Date   11 Sep 2000
Death Place   Greensboro, North Carolina
Occupation   Usher
Event Type   Obituary
Event Date   12 Sep 2000
Event Place   North Carolina, United States
Event Place (Original)   North Carolina, United States
Newspaper   Greensboro News & Record
Affiliate Name   GenealogyBank, Inc.

wife Doris and children's names are also listed

https://www.familysearch.org/ark:/61903/1:1:QVY3-SSLW?lang=en

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United States, Social Security Numerical Identification Files (NUMIDENT), 1936-2007
Name   Ernest Hansen
Sex   Male
Birth Date   7 Jun 1926
Birthplace   Brooklyn, Kings, New York, United States
Social Program Application Date   Oct 1942
Residence Place   Greensboro, Guilford, North Carolina, United States
Death Date   11 Sep 2000
Race   White
Father's Name   Gustav Hansen
Father's Sex   Male
Mother's Name   Hanna Christiansen
Mother's Sex   Female
Additional Person's Name   Ernest Hansen
Event Type   Social Program Correspondence

https://www.familysearch.org/ark:/61903/1:1:6KML-8L2F?lang=en

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Doris' date of death, 2017, on headstone
https://www.findagrave.com/memorial/195593961/ernest-hansen

Let's try working backward from your Alan Frost.
He doesn't look 54 in the picture taken in Toronto in 1958, I'd say 30 at most, so born between maybe 1918 and 1928.
For one reason or another, that might not be his birth name. I don't think government policy would have been a factor that recently, but it really doesn't affect the search.

So, do you have any DNA links to the Withington, Lancashire, area, where their Alan Davenport Frost was from?
Where did you test the DNA? Do you only have the general region, or do you match specific individuals?
If you have matched specific individuals, can you trace back their families?
Do you match anyone but the first cousin you mentioned? If you have a few good matches, sometimes you can "triangulate" to the common ancestor.
Are you or your cousin a direct male descendant of your Alan Frost? Have you submitted your y-DNA anywhere?
Where have you posted the DNA results? Try uploading them to a few sites, you might find more matches. Ancestry will not accept uploads, but if you tested there you can download the results and upload them to gedmatch or familytreeDNA, both of which have helped me and someone else I know. I think myheritage.com also accepts uploads.
Is the match in Rio helpful? Have you talked to them? Even if not, can you work backward from them?

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added: did you know this first cousin before hand? if not, how do you connect to him?

Your Alan Frost is not that far back. If he had children, brothers, sisters,  or 1st cousins, you should be able to find something at least.