Is there a word for a 60th of a second?

You could call it a "third" if you wanted to. But, of course, it is not commonly used. Etymologically:

pars minuta prima, first diminished part (1/60 of an hour), was shortened to "minute"

pars minuta secunda, second diminished part (1/60 of that), was shortened to "second"

So how would you shorten the next in line, 1/60 of that?

Here we go:

In 1267, the medieval scientist Roger Bacon stated the times of full moons as a number of hours, minutes, seconds, thirds, and fourths (horae, minuta, secunda, tertia, and quarta) after noon on specified calendar dates.

[Wikipedia]

added
In English, the Oxford English Dictionary has this as its earliest usage of third for angles: "Euery degree..doth containe 60 minutes, and euery minute 60 seconds, and euery second 60 thirds, &c." quoting Seamans Secrets (1595) by J. Davis

The term jiffy was used on the Commodore 64 and the Vic 20 to stand for 1/60th of a second (although Wikipedia claims it was either 59.94 Hz or 50 Hz).

Tick has been used for the same purpose (basic unit of time) on other computers.

The Wiktionary page for jiffy shows the meaning as "A unit of time defined by the frequency of its basic timer; historically, and by convention, 0.01 seconds, but some operating systems use other values", and Wiktionary defines tick as "A jiffy (unit of time defined by basic timer frequency)".

In the U.S., one could reasonably use the "electronics" sense of jiffy, which is "The time between alternating current power cycles (1/60 or 1/50 of a second)", as a name for 1/60th of a second.

My best bet is with Jiffy, which is an informal term.

In electronics (electrical engineering), a jiffy is the time between alternating current power cycles,[2] 1/60 or 1/50 of a second in most building power supplies — see alternating current.