Andrew G. Haley, who was von Kármán's lawyer at the time Aerojet was founded and was later president of the company. In his time at Aerojet, he became interested in space law, and in the late '50s he proposed the Kármán Line as a boundary past which national airspace jurisdiction should cease. He very loosly based his definition of the line on a paper by von Kármán on aerothermal heating and lift in high speed, high altitude flight and later named it the Kármán line.
In his autobiography, von Kármán takes credit for thinking of the point at which aerodynamic flight was no longer viable as a jurisdictional threshold, but so far as I know it is not mentioned in any of his notes or other writings. I went down a rabbit hole about this once and found nothing that connected von Kármán to the jurisdictional concept of the Kármán line until the autobiography was published years after Haley had coined the term and - presumably - the concept. I did come across a very interesting paper in the Journal of Space Law that sheds light on the subject:
The Non Kármán Line: An Urban Legend of the Space Age by Thomas Gangale
(That article is now helpfuly linked from the Wikipedia page on the Kármán line.)
That rabbit hole I went down started with trying to figure out why people have the the idea of space magically starting at 100km simply because the FAI and others say so (same would for for the US's definition of 50 miles). No airplane can fly remotely near that altitude on lift alone, yet it's far too low for a sustainable orbit. From an aerospace perspective, it's a useless concept. Jurisdictionally it makes more sense but is still arbitrary.
Re. the FAI's definition, it appears in
section 2.7.2 of their sporting code for astronautics. They refer to it as the "Von Karman ellipsoid," in deference to the shape of the Earth. Edit to add,
back in 2018, the FAI considered lowering their definition to 80km. I don't think anything has come from that yet.