From the word ayin we get ein עין and maayan מעין - meaning "spring, fountain", and according to Klein and others this is a shorter version of עין המים - "eye of the water".
Klein says that the word maon מעון - residence is related to ein, but I don't exactly understand how.
Here's one understanding of "how" these ideas are connected.
HaMasovevet can cast a ritual circle to create a Divine Space without using any tools at all. She can simply use her fingers.
To cast a ritual circle, she:
- chooses the place
- draws up energy from the feminine core of Shechinah within her
- extends the middle and index fingers of whichever hand she prefers, and draws that core energy into them
- encircles the space, marking its boundaries with the core energy flowing through her fingers
How is this finger-magick related to the excerpt on ayin? According to the Ari's system of chiromancy , the tips of the middle and index fingers of the right hand correspond to the hebrew letters reish ר and shin ש (resha d'ayin). The tips of the middle and index fingers of the left hand have no correspondence (resha d'lo itayada). There is not anything "written" on them.
The shoresh  אין refers to this left-handed "absence" of anything as yet written. It is the"place" made for the "no other" (of Ein Sof אין סוף). The shoresh also implies "from where" and/or a question of "location". In other words, אין is unknowably yet integrally connected to emunah (אמונה), the power of the soul through which all emanates to create "the where" of a "living" Divine Space. A Ma'on. Made through actions of the left hand.
R' Yitzchak Ginsburgh of the Inner Dimension teaches regarding Ma'on:
Ma'on ("the Residence")--our experience of the Divine forces that act in nature, enabling us to emulate God in our lives.
By casting a sacred circle with her left middle and index fingers, HaMasovevet taps into those Divine forces acting in nature to create a living Divine Space. Within this Sacred Circle - where the Divine, the spiritual and the physical are united - she is enabled to emulate God in her life.
With respect to the phrase "eye of the water" עין המים, the two fingers ע of the left hand י used to draw up ן the energy with which to create the Divine Space correspond to the "eye". The action of her hand המי corresponds to "of the water". The phrase ends with a mem-sofit ם. In both Rashi and cursive scripts, ם is written as a circle with a little "tail"  into אין סוף Ein Sof. We can see then, that the whole phrase can be interpreted as a description for making the Divine Space of a sacred circle.
 Meditation And Kabbalah, R' Aryeh Kaplan, p. 205
 Etymology Of Biblical Hebrew, R' Matityahu Clark, p. 9
 "A star [kochav] shall go forth from Jacob and a staff [shevet] shall arise in Israel." BaMidbar 24:17 - from the Inner Dimension: "As is well known, a comet has a "tail" that trails it (the Rambam asserts this fact in his laws of blessings). This is the secret of "the footsteps of the Mashiach"--the "traces" that are left by the King Mashiach on his way to redeem us."