I don’t remember if the story specified that there would be exactly 9,000,000,000 names decimal. It seems more likely to me that it would be some number about nine billion reflecting some value arising from combinatorics; which in fact the story was about how a computer program outputting to a printer could methodically work through all the combinations. I don’t think it would be 233 ,that is too close to exactly between eight and nine billion, and 234 is way too big. So any plausible “Tower of Hanoi” functions that give something ≥ 9.0 billion and, say, <9.2 billion?
Clarke gave himself a fudge factor - certain combinations of letters (the same letter three times in a row was one example the chief monk mentioned) are not allowed, and thus don’t count against the 9 billion.
Did Clarke mention that different languages have different alphabets containing different amounts of letters?
IIRC it was implied it was Tibetan(?) so not sure what language/alphabet they’d be using. Tibet uses something called a Brahmic script? Plus it has vowel markers for consonants, according to wikipedia. Seems complex.
Imagine monks speaking and working with Rotokas, a language with only 12 letters spoken by about 4,000 people.
Okay, I reread the story. It says “about” nine billion names. The monks say that they are using their own custom alphabet so it’s not a logographic script like Chinese, and no name can be longer than nine letters (but presumably some are shorter). But we’re not told all of the disallowed permutations. Naively all one can say is that their alphabet must have at least thirteen unique characters.
Too late to edit: no, I got that wrong; it could well be less than thirteen characters when you add in less-than-nine letter names.
Suppose your alphabet has m letters. The number of strings of length k equals m^k. Therefore the total number of strings of length between 1 and 9 is (m^{10}-m)/(m-1).
P.S. tables of “names of God” of various letters occur in Jewish mysticism, etc.
The .pdf I found online includes Clarke’s preface, which includes:
Its basic arithmetic was later challenged by J.B.S. Haldane, but I managed to save the situation by alphanumeric evasions whose precise nature now escapes me.
Now I wish I knew what Haldane had to say about it.
ETA: just reread the story again and caught a detail I hadn’t realized before:
somewhere among all the possible combinations of letters that can occur are what one may call the real names of God. By systematic permutation, we have been trying to list them all.
IOW, the monks are certain that if they list all possible permutations of their custom alphabet, all the true names of God will be included in that list. The text is ambiguous over whether that means nine billion permutations, or nine billion true names buried in a larger list.
they believe that when they have listed all His names—and they reckon that there are about nine billion of them—
You can put some bounds on the number of permutations they’re going through from the amount of time they were running the program, the sort of printer they were using, and the power output of the generator they’re using. IIRC, the time and power were explicitly given in the story, and the style of printer can be inferred from the sound they’re described as making.
An original-model IBM 1403 (future tech when that story came out, but this is science fiction, right?) could print 600 lines per minute. Seems pretty slow if you are going to print one name per line. However, there are 120 columns (I think?), so if you squeeze in 12 names per line it seems like it would only take a couple of years to print all that out, divided by the number of printers.
Right, 9 billion fits fairly comfortably. But as @Lumpy points out, there’s the possibility that 9 billion is just the number of the True Names, buried in amongst a larger set of candidate-names. If so, one can bound the size of that larger set from the printer capability.
OK, digging up my copy: From the story, the project will take a hundred days or three months (both figures obviously rounded), and the generator is fifty kilowatts, and is also powering prayer wheels plus unspecified other amenities that make the monastery much more comfortable. The machine is also described as producing an “incessant patter, the never-ending rainstorm of the keys hitting the paper”, which I think is consistent with a line printer like the 1403. But I can’t easily find figures for its power consumption.
There’s also a mention of the computer spending “weeks now” churning out “acres of pages”, but I don’t think this is precise enough to do much with.
I thought the same letter three times or was it four times in a row was permitted and one of the computer experts brought in to speed up the process questioned why such a high number.
I can’t access it online and I don’t have a physical copy so if someone can/does, please check the text for me. It’s now bugging me that I don’t remember that point.
I was mistaken - it’s no more than three consecutive identical letters are allowed
When the lid to our line printer “fell off’ (an operator was annoyed at it and opened it too hard) we had to wear ear protection in the room. More than patter. 
A line printer would be 120 characters by 66 lines (no page break space) and if we squeeze 12 names on a line, that’s still 11M pages. Google suggests a box (a little over a foot deep, say 14”) contained 2700 pages, so 4200 boxes of paper - 15”x11”x14”. And they were IIRC from the story decades ago, leaving by horse on a mountain trail or something. …logistics.