The word Rishi (Great Yogi, Phra Ruesi, Lao: ພະລືສີ, Thai: พระฤาษี) may be derived from two different meanings of the root 'rsh'. Sanskrit grammarians derive this word from the second meaning: "to go, to move". V. S. Apte gives this particular meaning and derivation, and Monier-Williams also gives the same, with some qualification.
Another form of this root means "to flow, to move near by flowing". (All the meanings and derivations cited above are based upon Sanskrit English Dictionary of Monier-Williams). Monier-Williams also quotes Tārānātha who compiled the great (Sanskrit-to-Sanskrit) dictionary named "ṛṣati jñānena saṃsāra-pāram" (i.e., "one who reaches beyond this mundane world by means of spiritual knowledge").
More than a century ago, Monier-Williams tentatively suggested a derivation from drś "to see". Monier-Wiliams also quotes the Hibernian (Irish) form arsan (a sage, a man old in wisdom) and arrach (old, ancient, aged) as related to rishi. Monier-Williams also conjectures that the root drish (to see) might have given rise to an obsolete root rish meaning "to see".
However, the root has a close Avestan cognate ərəšiš "an ecstatic" (see also Yurodivy, Vates). Yet the Indo-European dictionary of Julius Pokorny connects the word to a PIE root *h3er-s meaning "rise, protrude", in the sense of "excellent" and thus cognate with Ṛta and right and Asha. In Sanskrit, forms of the root rish become arsh- in many words, (e.g., arsh)
Modern etymological explanations such as by Manfred Mayrhofer in his Etymological Dictionary leave the case open, and do not prefer a connection to ṛṣ "pour, flow" (PIE *h1ers), rather one with German rasen "to be ecstatic, be in a different state of mind" (and perhaps Lithuanian aršus).