Web searches mostly turn up various copies and translations of the Wikipedia article. The WP article is useful because it clarifies the fact that 5-6 is not the only number he used, nor did he always use numbers. He also used 13 and 101 to construct names (13 multiple times), and some of his names are foreign words that aren't numbers, such as Jestocost. So this evidence makes it seem like there is probably not any overriding significance to 5-6, as if it were a Dan Brown key to unlock the hidden meaning of everything he wrote.
I don't have my copy of The Dead Lady of Clown Town to hand, but there is a 2008 article by Carol McGuirk, "Science Fiction's Renegade Becomings," that seems to shed at least some light on this. The article appeared in Science Fiction Studies Vol. 35, No. 2, On Animals and Science Fiction (Jul., 2008), pp. 281-307 (27 pages), but there is also a non-paywalled version here. She says:
The story opens with the conception of Elaine at An-fang (German word
for beginning), a place near Meeya Meefla, as Miami, Florida, is
called in Smith’s far-future. Due to a malfunctioning computer and the
distraction of a young human overseer, a human child is genetically
coded to become a “witch-woman” or, as the computer puts it, “lay
therapist, female, intuitive capacity for correction of human
physiology with local resources” (223). Most “witch-women” are sent to
frontier worlds, but it will be this child’s fate to grow up on
Fomalhaut III, a planet that requires little in the way of intuitive
healing, since its uniformly prosperous people are seldom anxious, let
alone sick. The same child is in error assigned an “animal” name,
Elaine: human beings in this period are said to be identified by
numbers.17 In a plot-twist reminiscent of Sturgeon’s sf, half-mad
Elaine, a “mistake” of a person, will play an important part in
breaking down rigid hierarchies and oppressive ways of thinking.
Footnote 17 says:
In “The Dead Lady of Clown Town,” Lady Goroke’s name suggests the
Japanese numbers for five-six (go and roku). Panc ashash suggests the
Hindi words for five-six (paanch chah), and femtiosex—Lord Femtiosex
is the harshest of D’joan’s judges—in Swedish means fifty-six. The
name Veesey-Koosie in “Think Blue, Count Two” (1962), sounds like
five-six in Finnish: viisi kuusi. While there are other examples,
number-names are not invariable. Jestocost, as mentioned, comes from
the Russian word for cruelty; and in “Alpha Ralpha Boulevard” (1961),
Menerima’s name (she becomes Virginia after the Rediscovery of Man),
although said in the story to be a number, in Indonesian means
“accept” or “swallow”—suggesting Virginia’s fatal indoctrination into
reflexive contempt for underpeople. Perhaps Menerima is a number in
some other language, however. Smith played with the meaning of words
from many cultures.
So the text itself does seem to provide at least a partial answer to your question. There is an explicit thematic/metaphorical idea that numbers and computers are about oppression, which is posed as the opposite of freedom and animalism. The idea, then, would be that it's significant (and explicit) that the names are numbers, and this has a point in the story. However, the reader is not expected to know what numbers they are based on.
So we have more than one number that gets used, some numbers are repeated and some are not, and the story/metaphor logic says that it doesn't actually matter very much which numbers they are.
It may be that 5-6, 13, and 101 had some personal significiance, or they were in-jokes, but if so, it doesn't sound like we'll ever know unless someone who knew him comes forward to say so. Most likely he just wanted cool-sounding names that would help to give a sense of wonder to his universe, and he found that generating them in this way worked well for that purpose. This is, after all, an old problem that SF authors have struggled with, often with bad results. I still cringe when I remember all the extruded-fantasy and extruded-SF books by Andre Norton that my public library had when I was a kid. All the characters had names like Jark and Blem.