It's not officially revealed how the designated survivor is chosen. They are always chosen from one of the individuals in the presidential line of succession who is constitutionally eligible to become President. Usually, one such Cabinet member is chosen for events such as State of the Union addresses and presidential inaugurations. For instance, Secretary of the Interior David Bernhardt was chosen as the designated survivor for the 2020 State of the Union address.
Note that this is a practice, but neither legally required by the Constitution nor by the Presidential Succession Act.
From the Senate Historian Office:
Each year, one member of the president's cabinet is absent from the address, to maintain the line of succession in case of an emergency.
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The practice of one cabinet official remaining absent from the event dates at least to the early 1960s and perhaps much earlier. Prior to the 1980s, however, the selection of the official was often not made public.
As ABC News explains:
The designated survivor is typically a Cabinet-level official who is chosen to skip certain high-profile events -- generally limited to State of the Union addresses and inaugurations -- so that he or she may assume power if a disaster were to occur while the majority of the government is gathered at the event.
It appears that the President makes the decision, according to the National Constitution Center.
Since then, a designated survivor has been used for the State of the Union, inaugurations, and presidential speeches to joint sessions of Congress. It is believed the President makes the decision of which Cabinet member is absent with permission from these events.
This article by Bustle backs up the claim that the President makes the decision on the choice of designated survivor and also sheds some light on that process.
CBS News reports that the president and their staff are responsible for selecting the designated survivor, and Jon Favreau, Barack Obama's former lead speechwriter, spoke to The Ringer about the designated survivor selection process in 2016. Favreau initially said that the process is "entirely random," but then backtracked a bit and said that sometimes, the designated survivor depends on what the president intends to say in their speech.
"Sometimes the designated survivor is chosen based on, 'are their programs or policies going to be a highlight of the State of the Union?'," Favreau said at the time. "I remember years where education would be a big deal in the speech and therefore Arne Duncan, who was the education secretary at the time, could not be the designated survivor."