Why did the IETF specifically choose 192.168/16 to be a private IP address class?

I know who chose these address ranges. Unfortunately, he is dead, so I cannot ask him exactly why he chose them, but I can make some well informed guesses.

There isn't much online dating prior to the mid-1990s, when the Internet really started to take off. What history of the Internet exists is mostly in the RFCs which define it, which date back to 1969, at the start of the ARPANET. Through them you can watch the Internet progress from a fledgling network of a few primitive mainframes, being designed by some of the most brilliant minds of the time, to the network we can hardly imagine living without today.

This answer draws almost entirely from those RFCs, and in small part from my personal experience as I was on the Internet in this era.

First, the IETF did not pick these IP address ranges, or any others. Allocation of special use addresses is currently and has always been the job of the Internet Assigned Numbers Authority.

The IANA has always been a role, rather than a specific organization, and that role has changed hands exactly once. Currently it is held by ICANN, but from 1972 until his death in 1998 when that organization was created to replace him, IANA was essentially one man, Jon Postel. Of course, he first called the role czar of socket numbers, a necessary task he took on himself because it needed to be done. He ended up the czar of virtually every number that could be assigned: addresses, protocol numbers, ports, you name it, largely because he was willing to do it, and by the time the Internet opened to public commerce he had been doing it for over 20 years. He assigned the numbers, and the Internet Registry (then SRI-NIC, this was expanded to a distributed collection of registries worldwide) published them.

The last RFC from SRI showing a list of Internet address assignments was RFC 1166 from 1990. It is a very lengthy list, so it should not be surprising that this data was moved to online databases. Comparing it to its predecessor RFC 1117 shows the rate of expansion of the Internet even then, years before it opened to the public.

So, now we are in a position to understand the address ranges in RFC 1918 a little better. This is actually the second revision of the RFC; the first was RFC 1597, published almost two years earlier in March 1994. In its little-known rebuttal, RFC 1627, the contemporary arguments against private address spaces were laid out. RFC 1627 also happens to mention who assigned the three address spaces.

They were assigned by IANA, that is, Jon Postel, at the request of the authors of RFC 1597, and if the complaint in RFC 1627 is to be believed, he did so via back channels rather than the usual open processes. You can see that RFC 1597 itself went straight to RFC status without the usual preceding Internet-Drafts, so it too was approved via back channels, again by Postel, who was also RFC editor at the time. So it might never be possible to answer this question conclusively.

Now as to why he chose these three address ranges, let me return your attention to the RFCs 1166 and 1117 from SRI which had the then-current IP address range assignments. In both of them you will notice that network 10 was still allocated to the defunct ARPANET, which had shut down in 1990. Postel, in his role as IANA, would know that this range was no longer in use and could be reassigned. I posit that Postel picked network 10 because he knew it to be available and not in use.

Similarly, I expect Postel picked 192.168 because, at the time he made the choice, it was the next available, or nearly the next available, network to be assigned from the former Class C space. This probably can't be proved one way or the other, but the pace of address assignments shown in the RFCs strongly suggests that they would have been in this general vicinity around 1993-1994 when the assignments were made. (Addresses in 192.159 were being assigned in 1992. No dates are available for assignments in 192.160-192.167 as these were at some point reallocated to RIPE.)

Answering this question for 172.16-172.31 is more difficult. Nothing I could find suggests why this range was selected. Assignments in the former Class B space had not gotten nearly that high yet, as far as I can discover. I can only speculate that IANA threw a dart at a dartboard, rolled dice, or otherwise pulled the number out of his nether regions.

Finally, a note about Jon Postel. Despite the apparent way in which this RFC was brought about fully formed without (initial) input from the community, I do not mean to imply that, and this should not be construed as, Jon Postel somehow executed the IANA role poorly or unfairly. He was one of the strongest influences on the early Internet, and you still feel that influence today every time you get a glimpse of the behind-the-scenes machinery of the Internet, but he was always concerned with doing the job right. To quote from one remembrance:

There is no glory in doing administration and operations. Quite the opposite. People notice when it is done badly but rarely offer praise when it is done well. People in administrative positions often become petty bureaucrats. Since there is so little reward in the job, they artificially make it a base of power. So it has confused some who heard Jon referred to as the Internet numbers "czar". They did not realize that the community imparted the title to Jon out of affection and deep appreciation for his having brought order to essential infrastructure services. In particular the community used that term in full knowledge that Jon took his position as a trust, rather than as an opportunity for personal power. We always knew that his views came from legitimate beliefs and we never had to worry that he was somehow considering political or personal advantage. We might not agree with him, but we always knew was driven first by a concern that the right thing be done.