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AHD-IER (Watkin, 2011) P93 gives PIE *tn̥-yā- for Gk. ταινία:

Suffixed zero-grade form *tn̥-yā‑. taenia; polytene, from Greek tainiā, band, ribbon.

while EDG (Robert Beekes, 2010) P1444:

Probably ultimately related to τείνω, but the details are rather unclear.

Is it a convicing derivative of PIE *ten- "to stretch"?

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    Semantically, the connection is clear enough; the same root leads to English thin. The phonological convincingness is dealt with in fdb's answer. – jlawler Aug 14 '18 at 15:46
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Semantically, the connection is clear: when something stretches across a long distance, it becomes quite thin. PIE *t-n- also underlies English "thin" via Germanic and "tenuous" (stretched almost to the breaking point), "tense" (stretched out), "tent" (canvas stretched over poles) via Latin via French.

The difficulty isn't in the semantics, but in the phonology. Proto-Indo-European *n̥y generally turns into /anj/ and later /ajn/ in Ancient Greek, and (as fdb points out) there isn't a solid explanation of where the extra /i/ in the second syllable could have come from: the known rules would predict /tajna/ ταίνα. These are the unclear details Beekes is referring to.

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    PIE *n̥ actually regularly gives Greek α. But before *y, it seems to have gone to *any and later αιν, as in βαίνω < *gʷm̥-yō. So the problem here is not the /ajn/, but the ι in the second syllable. – TKR Aug 14 '18 at 19:17
  • @TKR Good call! Fixed. – Draconis Aug 14 '18 at 19:48
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The difficulty with the derivation of ταινία from *tn̥-yā- is the diphthong in the first syllable. This is why Beekes calls it "rather unclear".

  • I think the difficulty is rather the vowel in the second syllable: *tn̥-yā should give ταινα, like *gʷm̥-yō > βαίνω. – TKR Aug 14 '18 at 19:09
  • The point seems to be that you should not have /j/ in both syllables. Once is enough. @TKR – fdb Aug 14 '18 at 19:12

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