"Spirit of holiness" corresponds to the exceptional semitism found in the Greek of Romans 1:4, pneuma hagiosunès, a calque of the Hebrew idiomatic expression found in Isaiah 63:10 (ruach qâdshô, literally, "the spirit of his holiness") or Psalm 51:13 (ruach qâdshekha, "the spirit of your holiness") -- although in those two instances the LXX uses the normal Greek adjectival form with the possessive applying to the entire expression, to pneuma tou hagion autou/sou = "his/your holy spirit".
Another important stepping stone for the NT evolution of the concept is the "personification" of "hypostatisation" (in angelic fashion) of the "Spirit of Truth" vs. the "Spirit of Error" in late Judaism (cf. Testament of Judah 20 and Qumran Rule 3:18ff) which is directly echoed in Johannine writings (especially 1 John 4:6, although this is lost to NWT readers).