This term might not be a reference to the Holy Spirit. Rather it could speak of the gift of faith that He imparts to us. However, it would certainly work as a name for the Holy Spirit because He imparts that faith to us.
Spirit of faith in Greek:
To pneuma tēs pisteōs
Strong’s Concordance numbers: 4151, 4102
Bible reference: 2 Cor. 4:13
We having the same spirit of faith, according as it is written, I believed, and therefore have I spoken; we also believe, and therefore speak. (2 Corinthians 4:13 KJV)