Greek ποτα-μός is the outcome of *ku̯oth₂-mó- ‘foamy, foaming’, a primary adjective with structure CoC-mó- (of the same type as Proto-Indo-European (PIE) *gʷʰor-mό- ‘warm’, attested by Vedic gharmá- ‘heat’, Latin formus ‘warm’) of the PIE root *ku̯eth₂- ‘foam, seethe’.
The root *ku̯eth₂- is attested (among others) by Vedic Sanskrit kváth-ant- ‘foaming, seething’ and Gothic ƕaþjan* ‘to foam, ἀφρίζειν’; the current reconstruction of the root as *ku̯ath₂- (LIV2: 374), with -a- vocalism, relies on the highly problematic connection with Latin cāseus ‘cheese’ and should be dropped.
The oxytone accentuation of ποταμός, consistent among Greek primary -mό- adjectives (such as θερμός ‘warm’, ὠμός ‘raw’, and δοχμός ‘slant’), speaks against an analysis as a denominative formation derived by means of a secondary suffix -αμoς from, for instance, a noun *πότ-ο- ‘foam’ (PIE *ku̯óth₂-o-) matching Proto-Germanic *hwaþ-a- ‘foam’ (which is reflected by Gothic ƕaþjan* ‘to foam’, ƕaþo ‘foam’, and Swedish kva ‘id.’).
The theonym Τηθῡ́ς is the reflex of *ku̯ēth₂-ú-h₂ ‘foamy-ness, seething-ness’, an abstract derivative of the type of Homeric ἰθῡ́ς ‘direction’ (: ἰθύς ‘straight’) and Vedic tanū́- ‘body’ (*‘length’ : tanú- ‘long’) of an adjective *ku̯ēth₂-ú- ‘foamy, foaming, seething’, which is attested by the Homeric hapax τήθεα ‘sea-squirts’ (Iliad 16.747), animals that violently expel water from their orifices, and by the Hesychian gloss τηθύα ‘lagoons at the mouths of rivers’. The lengthened grade of the root may be due to the influence of a Narten present with ablaut *ku̯ḗth₂-/ku̯éth₂-; alternatively, *ku̯ēth₂-ú- may be analysed as a derivative of the type of Homeric ἤνις ‘of one year, one-year-old (of cows)’ (: ἐνι° in, e.g., ἐνι-αυτός ‘anniversary, lapse of a year’) of the weak stem of *ku̯óth₂-u-/ku̯éth₂-u- ‘(state of) foaming, seething’.
The semantic development from *ku̯oth₂mó- ‘foamy, foaming, seething’ to ποταμός ‘river’ and from *ku̯ēth₂-ú-h₂- ‘foamy-ness, seething ness’ to Τηθῡ́ς, name of the spouse of Ocean and mother of all rivers, reflects the traditional association of rivers, ocean, and bodies of water in general with foaming and seething, attested in the phraseology of Greek itself (for instance, by the Homeric formulaic expression ἀφρῷ μορμύρων ‘seething with foam’, which always refers to rivers and to Ocean) and of other Indo-European traditions, such as Latin (see Vergil and Lucan’s formula spumeus amnis "foamy river"), Vedic (see RV 9.86.43c síndhor ucchvāsé “in the bubbling up of the river”), and Old Norse (see Lausavísur from Magnúss saga berfœtts 6.1-2 viðr þolir nauð í lauðri “the timber [= ship] suffers distress in the foam [= sea]).” The association of PIE *ku̯eth₂‑ ‘foam, seethe’ with the ocean finds further support in Old English phraseology, as another reflex of *ku̯eth₂- is the verb hwaþerian/hwoþerian ‘to foam, seethe, roar’, whose usual subject is precisely the sea (see ÆCHom II, 28 […] Se brym hwoðerode under his fotswaðum […] “the sea roared under his footsteps”).