The ... operators are identical to the range operator (..) in list context and nearly identical to the flip-flop operator (..) in scalar context, but calling them the range operator and the flip-flop operator seems wrong since those names are more commonly associated with .., which has slightly different behavior (in scalar context at least).
For now, I am calling them the alternate range/flip-flop operator.
I like to think of it as the ellipses operator--which makes it clear that it's about multiple dots (".." or "...") and less confusion about it's function.
In 5.11, where a term is expected (due to a bug, currently only at the beginning of a statement), ... is the yada yada operator.
Otherwise, in list context, ... is the range operator (though I would regard it as code smell, since the code seems to be wanting something different than .. but isn't in fact any different).
Otherwise, it is the flip-flop operator, one flavor thereof. If I had to give it an adjective, I would say the sed-like flip-flop operator. In the perl6 spec it (well, the fff replacement, anyway) is called "flipflop (sed style)". If I wanted to give it a name not based on another language, I'd start by getting the perl6 spec updated, then update the perl5 doc.
Since ... is identical to .. in list context I'd call it the same thing: the range operator. Giving it another name would imply that it does something different. If I needed to distinguish it from .. for some reason I'd probably call it the "three-dot syntax for the range operator."
If I wanted to mess with people I'd tell them that it's "for really long ranges." ;)
In scalar context I've generally called ... the "sed-like flip-flop operator" because of the reference to sed behavior in the documentation, but I don't like that for a name. How about the "long flip-flop" operator? The mnemonic is that ... is one dot longer and takes one more cycle to evaluate the right operand.