The phrase that the NASB translates as "inner man" is ἔσω ἄνθρωπος (esō anthrōpos). Paul uses the same expression in Ephesians 3:16:
That He would grant you, according to the riches of His glory, to be strengthened with power through His Spirit in the inner man
ἵνα δῷ ὑμῖν κατὰ τὸ πλοῦτος τῆς δόξης αὐτοῦ δυνάμει κραταιωθῆναι διὰ τοῦ πνεύματος αὐτοῦ εἰς τὸν ἔσω ἄνθρωπον
(The phrase is also understood in 2 Corinthians 4:16 - see below.)
The word the NASB translates as "mind" is νοῦς (nous). The word appears 24 times in the New Testament, and the NASB uses the word "mind" on all but 3 occasions ("comprehension", Phil 4:7; "composure", 2 Thess 2:2; "understanding", Rev 13:18).
Another Greek word for "mind" is διάνοια (dianoia), which has νοῦς as its root. It is the word that appears, for example, in You shall love the lord with all your ... mind (Matt 22:37, Mark 12:30, Luke 10:27).
The inner man is not the same thing as the mind. By law of my mind, Paul is referring to the conscience, which struggles against the law of flesh (law in my members).
John of Damascus (676-749) contrasted the two laws in Exact Exposition of the Orthodox Faith (IV.22):
Therefore the law of my mind, that is, the conscience, sympathises with the law of God, that is, the precept, and makes that its will. But the law of sin, that is to say, the assault
made through the law that is in our members, or through the lust and inclination and movement of the body and of the irrational part of the soul, is in opposition to the law of my mind, that is to conscience, and takes me captive (even though I make the law of God my will and set my love on it, and make not sin my will), by reason of commixture: and through the softness of pleasure and the lust of the body and of the irrational part of the soul, as I said, it leads me astray and induces me to become the servant of sin.
The mind, in the context of Romans 7:22ff - is not itself the inner man, but is rather a part of the inner man. The law of the mind - the conscience - is a function of the mind. The inner man, as the Ephesians passage implies, is something deeper - wherein the Holy Spirit may dwell. The distinction is further drawn in 2 Corinthians 4:16:
Therefore we do not lose heart, but though our outer man is decaying, yet our inner man is being renewed day by day.
Thus, the inner man is the rational side of man, seen in opposition to the outer man - his mortal body.