I strongly suggest everyone read the original passage in the link. This quote makes it seem like it comes out of nowhere.
While I won't speculate on the specifics of why this was funny (you'd probably need to know more about press secretary Sheila Weidenfeld's role in the Ford administration) it's clear this was simply a joke of some kind on the final day of of Ford's presidency from the full context.
Mrs. Ford decided to break the tension. She looked over at Sheila Weidenfeld with a mischievous gleam in her eye and cried: “I’ve got some unfinished business to take care of!
"Break the tension" "Mischievous grin" "unfinished business" last day in the white house; it all makes it clear this was a joke that everyone in the room understood.
Maybe someone who knows more about Ford can elaborate on the role of the press secretary?
I hope this works as a response; I'm not very well versed in American politics, but I'm at least familiar with the gendered aspect of this, so... :)
Right off the bat, all the sources I can find on Betty Ford and spanking lead back to commentary on how she did not believe in spanking her children i.e. Jeffrey S. Ashley's Betty Ford: A Symbol of Strength states on page 52 that Ford did not believe in spanking her own children. The post linked in OP credits Paul F. Boller, who credits Press Secretary Sheila Weidenfeld herself, in First Lady’s Lady. I cannot find a digital copy of this book, nor can I find any other sources save for Weidenfeld herself that corroborates this event, which is a little unusual. The passage also refers to Weidenfeld in the third person, which seem a little odd for someone who was not only present, but also presumably being spanked.
However, that aside, the concept absolutely and completely speaks to the "popularity" of spanking in the mid 20th century, which had waned a little bit before Ford's time as First Lady. The spanking of adult women was seen as a way to "correct" unwanted behaviors, particularly in women who flouted gender roles or were seen as too emotional, flighty, disobedient, etc, and it appeared in a truly absurd amount of American cinema contemporary to the time. In fact, the way this interaction is framed sounds a lot like popular culture depictions of spanking:
It plays out just a bit too cleanly, doesn't it? But this isn't to say that Weidenfeld lied or made up the interaction –– spanking and "domestic discipline" were absolutely real issues, subject to a great deal of scrutiny by the legal system for its proliferation in actual, real-life married couples and sometimes even in places of employment. It's not entirely beyond the realm of possibility.
Andrew Heisel published a great in-depth layman's introduction to the topic of spanking adult women earlier this year entitled 'I Don't Know Whether to Kiss You Or Spank You': A Half Century of Fear of An Unspanked Women. Perhaps not a typical source here, but perhaps useful at the mods' discretion.
Hiya, boys and girls!
So, it's Christmas here on /r/AskHistorians, and I have a present for you all!
But really; our rules do still apply here, and that means that even for writing a response about... whatever the hell is going on here, we require answers to be in-depth and expert (no, you can't ask me what an expert on Lady Ford spanking the Press Secretary looks like), and we ask that everyone refrain from wild guesses, speculation, and shitposts other unhelpful comments.
Thanks for your understanding, and have a Merry Christmas. :)
^^I'm ^^Australian ^^it's ^^already ^^Christmas ^^here ^^so ^^I ^^can ^^say ^^that.