Etymology tree
Proto-Indo-European *tód
Proto-Germanic *þod
Proto-Germanic *þat
Proto-West Germanic *þat
Old English þæt
Middle English that
English that
English -'s
English that's
From that + -'s (possessive marker).
Meanings
(Canada, Northern-Ireland, US, West-Midlands) whose, of which (in dialects that require a human antecedent for 'whose')
1995, Aimo Seppänen & Göran Kjellmer. The dog that's leg was run over: On the genitive of the relative pronoun. English Studies 76, 389–400.
It had products like a WiFi light bulb that's brightness and even color can be controlled via Smartphone. 2018, Jimmy Im, "I shopped at Amazon's new 4-star-product store in New York City — here's why I wouldn't do it again", CNBC.com, 2018 September 28
"[…] we want them to bring a product to market that's time had not yet come," said Ray Farhung, a Southern California Edison official. 1993 January 10, Bill Vlasic, “Cool Contest”, in The Detroit News, page 1D:
that's is perfectly possible in my Northern Irish English dialect, e.g. "The cow that's calf was lost" is fine; "The cow whose calf was lost" is not so good as my dialect seems to require human reference for who, […] 2014 March 28, Warren Maguire, “['That's'] Comments”, in Mark Liberman, Language Log, archived from the original on 08 Apr 2014:
He had in his pocket an old flick-knife that's spring was slow and temperamental. 1992, Jim Crace, Arcadia, page 10:
[…] the only one that's title has been released […] 2009, Doug Whitman, quoted in Neal Whitman, "We Don't Speak the Same Language" (blog post, 2011 March 23)