Although on page 124 / Section 1.3.1 of Volume 1 of TAOCP, Knuth only gives the numbers (360+650+…), recall that his indexing is exhaustive. Accordingly, searching for page 124 in the index (easier with the PDF edition) gives all 16 computers whose model numbers are averaged. In alphabetical order, these index entries are:
Bendix G20, Burroughs B220, CDC 1604 computer, Honeywell H800, IBM 650 computer, IBM 709 computer, IBM 7070 computer, PDP-4 computer, Philco S2000 computer, RCA 601 computer (different from the Minivac 601 mentioned in the other answer), Recomp II computer, System/360 computers, UNIVAC III computer, UNIVAC SS80 computer, UNIVAC 1107 computer, XDS 920 computer.
That's the answer, but a few other remarks tangentially, some probably well-known to regulars at this site:
When TAOCP was conceived of in 1962, there was really a need for such a "mix" of computers to teach with, as there was a wide variety in computer architectures at the time, and programming directly for a machine was often what programmers did. For instance, many computers (including several in the above list) were decimal (rather than binary) computers. You can see this in the text just before section 1.3.1, and note how it differs between the first edition of 1968 and the current (third) edition of 1997 (bolding added by me, to highlight some of the diffs):
There should be no hesitation about learning a new machine language; indeed, the author has found it not uncommon to be writing programs in a half dozen different machine languages during the same week! Everyone with more than a casual interest in computers will probably get to know several different machine languages in the course of his lifetime. MIX has been specially designed to be so much like most existing machine languages that its characteristics are easy to assimilate.
versus
There should be no hesitation about learning a machine language; indeed, the author once found it not uncommon to be writing programs in a half dozen different machine languages during the same week! Everyone with more than a casual interest in computers will probably get to know at least one machine language sooner or later. MIX has been specially designed to preserve the simplest aspects of historic computers, so that its characteristics are easy to assimilate.
Several of the computers in the above list were of personal relevance to Knuth:
The IBM 650 was the first computer he encountered, and the TAOCP series of books is dedicated to the IBM 650. ("This series of books is affectionately dedicated to the Type 650 computer once installed at Case Institute of Technology, in remembrance of many pleasant evenings.") He also wrote a beautiful tribute to the IBM 650 ("The IBM 650: An Appreciation from the Field").
The Burroughs 220 is mentioned in the above tribute. ("Then I graduated, and began to tackle other machines. My favorite computer for the next five years became the Burroughs 220, which was another joy to use.") He wrote an Algol 58 compiler for the related Burroughs 205 machine (see "The Summer Of 1960 (Time Spent with don knuth)"), and was for a few years a consultant for Burroughs in Pasadena. (While he was a graduate student at Caltech, also in Pasadena.)
The Univac Solid State (SS80 above) was the third computer he wrote a compiler for, this time a Fortran compiler. From the Computer History Museum archive of several Knuth papers, see several documents that mention "Univac Solid State", e.g. this one. (See also this.)
One can guess from typical Knuth sense of humour that probably the name "MIX" was chosen as it was simply a "mix" of several machine languages, and then he noticed that "MIX" is also 1009 in Roman numerals, and made a joke out of it by choosing several computers whose model numbers averaged come out to 1009. (Rather than starting with 16 computers and averaging their numbers, and the rounded-down result in Roman numerals coincidentally being an appropriate word.) You shouldn't take this averaging too seriously, as the "origin story". He plays a similar trick elsewhere: although he once wrote, about his choice of "WEB" for his literate programming system, that
I chose the name WEB partly because it was one of the few three-letter words of English that hadn't already been applied to computers.
(he was the original WEB developer!), he says in the WEB manual that it is named after his mother-in-law (whose initials coincidentally happen to be WEB).
For his new computer "MMIX", he carries out a similar exercise in "Volume 1, Fascicle 1: MMIX", and also mentioned in some talk the effort it took to retain the joke by finding computers averaging to 2009, this time without rounding:
(Cray I + IBM 801 + RISC II + Clipper C300 + AMD 29K + Motorola 88K + IBM 601 + Intel i960 + Alpha 21164 + POWER 2 + MIPS R4000 + Hitachi SuperH4 + StrongARM 110 + Sparc 64)/14 = 28126/14 = 2009.