1. Serbian (Savić): patronymic from the personal name Sava, from Greek Sabbas (see Savas).
2. Jewish (eastern Ashkenazic): habitational name for someone from a place called Savichi in Belarus. Compare Savitsky.

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1. Greek: from the personal name Sav(v)as, New Testament Greek Sabbas, a derivative of Sabbaton ‘Sabbath’, ‘Saturday’. This name was borne by several saints and martyrs venerated in the Eastern Church, in particular a hermit (439–532) in Palestine, prominent in theological debates of his day.

source: Ancestry.com