What year did the seat belt sign come in to use?
The earliest reference to seat belt signs I can find is Scheduled Air Carrier Rules, 14 C.F.R. (1941)

§ 61.342 Seat belt sign. An aircraft shall not be operated in
scheduled air transportation unless a suitable means for warning
passengers to fasten seat belts is provided. [As added by Amdt. 129,
CAR, Sept. 5, 1941, effective Oct. 1, 1941, and Amdt. 130, CAR, Sept.
12, 1941; 6 P.R. 4691, 4753]

https://www.loc.gov/item/cfr1943001-T14CIP61/
The amendment appears in 6 FR 4753
https://www.loc.gov/item/fr006181/
It's unclear where the motivation for this regulation came from, but it is likely mentioned in the Civil Aeronautics Board's meeting minutes for September 5, 1941. Unfortunately it does not appear that these minutes have been digitized. The relevant records should be available in the National Archives RG 197.2.