"Holy Spirit" is a term used commonly in early Christian literature as well as some Second Temple Jewish sources, but it is rare in the Hebrew Bible. How rare? Well, the phrase [רוח קדשׁו] appears a total of three times in two passages! It can be found in Psalm 51:11 and Isaiah 63:10-11.
In Psalm 51:10-11 the psalmist writes (NRSV), "Create in me a clean heart, O God, and put a new and right spirit within me. Do not cast me away from your presence, and do not take your holy spirit from me." The "new and right spirit" or "renewed spirit" [ורוח נכון חדשׁ] of v. 10b parallels the "holy spirit" [ורוח קדשׁך] of v. 11b. A renewed spirit is anthropological in the sense that it parallels a "clean heart". To receive a purified heart and a renewed spirit is a poetic way of speaking of a complete washing and regenerating of a sinful human. In v. 11 the psalmist's plea to remain in God's presence is paralleled with the plea for God's holy spirit to remain with the psalmist. On one hand, this is about presence in v. 11a and v. 11b: the parallelism suggests God removing holy spirit is the active equivalent to the passive action of being casts from God's presence. On the other hand, there may be a plea for one's life to be spared when this psalm is read in light of psalms 104:29-30 and 143:7 where the removal of God's presence is the removal of the spirit-breathe that sustains human life.
In Isaiah 63:10-11 we have "holy spirit" used in closed proximity. In v. 10 the prophet says that Israel "rebelled and grieved his [God's] holy spirit" prior to being exiled. In v. 11 the prophet asks "where is the one who put within them his holy spirit." In both verses we read "his holy spirit [רוח קדשׁו]" though the nuance is slightly different. In v. 10 holy spirit seems to be equivalent to when a human says their "spirit is grieved" or something to that extent. It is God's "inward" emotional reality and it is holy. In v. 11 holy spirit seems to refer more to presence. This isn't to say it has nothing to do with God's person as is the case in v. 10, but it does have a nuance of dwelling or abiding with the people.
If our only sources for pre-Christian pneumatology was the Hebrew Bible we'd suggest that "holy spirit" meant one or more of the following: God's presence, God's sustaining spirit-breath, or God's own "inner" emotional self.