BQ: Some people will say that the indwelling of the Holy Spirit is simply "the spirit," meaning that of man, and not the Spirit of God. Is there any evidence that this is false?
Well yep, there's a lot of evidence. In the Greek, the word "pneuma" can mean both breath and spirit. Often the translators made it "Holy Spirit" based on context. However, some places explicitly modify the phraseology to specifically mean the spirit of deity. For example, 2 Tim 1:14 says, "Guard, through the Holy Spirit who dwells in us, the treasure which has been entrusted to you." This isn't a generic penuma/breath/spirit, but it is modified by "hagios," meaning, "sacred, holy."
A good question is, "Or do you not know that your body is a temple of the Holy Spirit who is in you, whom you have from God, and that you are not your own?" (1 Cor 6:19)
BQ: Yesterday we saw some evidence that a "holy spirit" inhabits Christians, but what if it just means, "sacred spirit," and it's just man's spirit that we need to consider sacred? Is there any other evidence that it's more than just the spirit of man?
You know that the answer is yes, huh? : P Two references we see are in the original texts as 'pneuma theos' and 'pneuma christos,' which translated mean, "Spirit of God" and "Spirit of Christ." Observe Romans 8:9-11
"However, you are not in the flesh but in the Spirit, if indeed the Spirit of God dwells in you. But if anyone does not have the Spirit of Christ, he does not belong to Him. If Christ is in you, though the body is dead because of sin, yet the spirit is alive because of righteousness. But if the Spirit of Him who raised Jesus from the dead dwells in you, He who raised Christ Jesus from the dead will also give life to your mortal bodies through His Spirit who dwells in you."