Telesinus is a habitational surname from Telesia (or Thelesia), and Telesinius is almost certainly a derivation of Telesinus. Telesia was a town founded in 217 BC by the Samnites near modern day Telese Terme in southern Italy, but I can't find any more authoritative information on the town's etymology.
My educated guess is that the name is from Greek given their contemporary influence in the region (for example, Samnites often used Greek words on their pottery). Based on the alternative spelling Thelesia, I suspect that it may come from Ancient Greek θηλεω ('theleo', "to bloom; to abound in"), thus meaning "flourishing; bloom; abundance." Alternatively, it could come from θελησις ('thelesis', "will, willingness; wish").
Sources:
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https://www.online-latin-dictionary.com/latin-english-dictionary.php?lemma=TELESINUS100 (Telesinus dictionary entry)
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https://it.wikipedia.org/wiki/Telese_Terme (modern town; in Italian)
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https://it.wikipedia.org/wiki/Telesia (ancient settlement; in Italian)
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https://en.wikipedia.org/wiki/Samnites (Samnite history)
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https://archive.org/details/schreveliusgreek00schr/page/n165/mode/2up (Ancient Greek dictionary)