If Euler introduced the term and did not explain his reasoning we can only speculate as to what he had in mind. Euler himself was followed on many notational and terminological choices simply because he put them together in well structured and comprehensive books.
But Euler likely followed the precedent with the "moment of force". According to Worthington's Dynamics of Rotation (1900), it came from a curious metamorphosis of the meaning of the word "moment" (originally, short duration), and the older meaning was still live at least at the time of Worthington's writing. According to EtymOnline, "moment" is used in old French as "importance" since the 12th century (in English since 1520s, think of "momentous"), such use in Latin must have come even earlier.
In On the Equilibrium of Planes Archimedes referred to the joint "importance" of force and the place where it was applied to maintaining the equilibrium. Commandino's Latin translation of Archimedes in Liber De Centro Gravitatis Solidorum (1565) was:"The center of gravity of each solid figure is that point within it, about which on all sides parts of equal moment stand". Now, one can read "moment" here as "importance", but one can also read it as referring to some specific quantity.
And that is, apparently, how it came to be read. "Moment" in physics now generally refers to quantities that are something times distance, and sums or integrals thereof. Higher moments involve the powers of distance. "Moment" in mathematics has a similar meaning, only applied more abstractly. Here is Worthington:
"The word moment was first used in Mechanics in its now rather old-fashioned
sense of 'importance' or 'consequence' and the moment of a
force about an axis meant the importance of the force with
respect to its power to generate in matter rotation about the
axis; and again, the moment of inertia of a body with respect
to an axis is a phrase invented to express the importance of
the inertia of the body when we endeavour to turn it about
the axis. When we say that the moment of a force about an
axis varies as the force, and as the distance of its line of action from the axis, we are not so much defining the phrase
'moment of a force' as expressing the result of experiments
made with a view to ascertaining the circumstances under
which forces are equivalent to each other as regards their
turning power. It is important that the student should bear
in mind this original meaning of the word, so that such
phrases as 'moment of a force' and 'moment of inertia' may
at once call up an idea instead of merely a quantity.
But the word 'moment' has also come to be used by analogy
in a purely technical sense, in such expressions as the 'moment of a mass about an axis' or 'the moment of an area with
respect to a plane', which require definition in each case. In
these instances there is not always any corresponding physical
idea, and such phrases stand, both historically and scientifically, on a different footing."