Why is it called commutative property? I can see why distributive is called distributive (distribute whatever you are multiplying to everything within the brackets).
Associative because when the same associative operator appear in a row, you can change around the numbers.
But why is the commutative property called commutative?
 A: https://www.merriam-webster.com/dictionary/commutative
https://www.merriam-webster.com/dictionary/commutation
Commutative, from commutation, which means exchange, trade, or replacement according to the first 2 definitions. The commutative property says that the order in which the operation is carried out does not matter. You can exchange/trace factors or addends and still arrive at the same product or sum. 
https://www.thefreedictionary.com/commutative
Definition one of the adjective form gives "involving substitution, interchange"
So you just switch or commute the two addends or factors and get the same sum or product!
A: From the Wikipedia article "Commutative Property", under History and Etymology:

The first recorded use of the term commutative was in a memoir by François Servois in 1814, which used the word commutatives when describing functions that have what is now called the commutative property. The word is a combination of the French word commuter meaning "to substitute or switch" and the suffix -ative meaning "tending to" so the word literally means "tending to substitute or switch." The term then appeared in English in 1838 in Duncan Farquharson Gregory's article entitled "On the real nature of symbolical algebra" published in 1840 in the Transactions of the Royal Society of Edinburgh.

A: Seems pretty reasonable since commute means to move around or change places, approximately.   One of the most basic ways of moving elements around is to switch their order.
I guess another word could have been used.  Of course "abelian" is used for groups, after Niels Henrik Abel. 
