The phrase mind of the Spirit is one of many similar phrases used to indicate a mind dominated by spiritual thinking as opposed to our natural way of thinking:
6 For the mind of the flesh is death, but the mind of the Spirit is life and peace;
Other phrases indicating the spiritual mind are:
Spirit of life in Christ Jesus:
2 For the law of the Spirit of life in Christ Jesus made me free from the law of sin and of death.
living according to the Spirit:
5 For those who live according to the flesh set their minds on the things of the flesh, but those who live according to the Spirit, the things of the Spirit.
Spirit of Christ:
9 But you are not in the flesh but in the Spirit, if it is so that the Spirit of God dwells in you. But if any man doesn’t have the Spirit of Christ, he is not his.
16 “For who has known the mind of the Lord, that he should instruct him?” But we have Christ’s mind.
walking by the Spirit:
16 But I say, walk by the Spirit, and you won’t fulfil the lust of the flesh.
fruit of the Spirit:
22 But the fruit of the Spirit is love, joy, peace, patience, kindness, goodness, faith,
Related theme: being led by the Spirit of God.