commutative group (which happens to be another name they often go by).
You're not about to guess what 'commutative group' is from it's name any more than you're going to guess what an 'abelian group' is, unless you have already learned what commutative or abelian means in the specific context of algebra.
I'm always baffled that programmers, who freely adopt frameworks and languages with mostly meaningless names (which incidentally makes them easier to google for), continue to insist that names are the major stumbling block in learning math. It's utter nonsense. You pick up the names and terminology quickly enough (or look them up if you forget); the hard part is applying them.
Naming things is important. Yes it just takes one googling, but in many cases you are interfacing with many different topics and just reading straight up the useful descriptive name would allow keeping the flow without having to look up stuff.
Maybe this is more of a stumbling block for some than others. Often I find I have to make up mnemonics and other strategies like "the longer word is the one that ..." or "put the two terms in alphabetical order to match their descriptive names' alphabetical order".
Again, this doesn't happen for terms we use every day. But if you use it every year or so, it's a stumbling block to have to ask "which one was that again?" when the descriptive name would immediately clear it up.
I think it's mostly vanity and a "respecting the elders" and credit assignment thing that so many things are named for mathematicians instead of descriptive names. Having something named after you is like one of the biggest "awards" a mathematician can get. But this has no concern for didactics.
https://dictionary.cambridge.org/dictionary/english/commute doesn’t give any definition close to the mathematical meaning, either.
Unless dictionaries 150-ish years ago had significantly closer descriptions of the term, “Abelian”, being a new term not loaded with pre-existing definition might be the better choice for naming this property.
"Order-independent" like I suggested communicates a lot.
Edit: “symmetric” might be an alternative for Abelian because the Cayley table (https://en.wikipedia.org/wiki/Cayley_table) of an Abelian group is symmetric. I’m not sure that’s immediately clear enough for laymen, though.
I suppose we could rename "symmetric group" to "maximal permutation group", but stringing adjectives together is not a sustainable strategy for naming. Plus, it is not clear to me how accurate "maximal permutation group" is as a name. That is, the symmetric group S_3 is the maximal permutation group of {1,2,3}. However, it is not generally a maximal permutation group, as it is contained withing the permutation group S_4.
Perhaps pair-independent is a distinctive name for associative...
“Pair invariant” IMO, isn’t good, certainly worse than “Pairing invariant”. “Parentheses invariant” might work, but of course would get just as confusing/incorrect as “pair invariant” once one moves from groups to fields.
Somehow including "peer" in the terminology could help.