1 Peter 1:10-11 "As to this salvation, the prophets who prophesied of the grace that would come to you made careful searches and inquiries, 11 seeking to know what person or time the Spirit of Christ within them was indicating as He predicted the sufferings of Christ and the glories to follow."
The significance of the Holy Spirit as "The Spirit of Jesus"
In the opening passage of today's post, the Apostle Peter indicates that the Holy Spirit's work in the Old Testament prophets aimed to prepare and point to the Son in His first coming. Although the prophets did not fully comprehend the full picture of what they were writing, Peter indicates that the Spirit's revelatory work was centered on revealing the Son. Hence as the "Spirit of Christ", we find the Holy Spirit being associated with the work of what would be and is the Person and work of Jesus the Messiah.
Paul in his letter to the Philippians speaks of the provision of "The Spirit of Jesus Christ" in Philippians 1:19. Here we are reminded of how the Holy Spirit delivers to the saints of God timely provisions which, in effect, are the ministrations of the Lord Jesus Christ to His people. When Paul indicated in Acts 16:7 that "The Spirit of Jesus" had hindered he and his companions from entering into Asia Minor, we find not only the Spirit being directly involved, but also the Son's involvement by association. The will of the Son is at work in the passage, which of course means that in ultimate terms, the will of the Father is working behind the scenes. Such language reminds us that all Three Persons of the Godhead work together as One God.
Other texts could be cited that utilize this title "Spirit of Jesus". Galatians 4:6 states that the Holy Spirit, sent by the Son, aids us in crying "Abba Father". The Spirit points not only to the Son, but also the Father. We could reference the other passages mentioned above, but this should suffice to show that the significance of the title "The Spirit of Jesus" and its variations aims to point the way to the Person and work of the Lord Jesus Christ, as well as the Will of the Father in the lives of His people.
As a final thought, The Holy Spirit ever represents the interests of the Trinity. It has been said that He is the Ambassador of the Trinity - and thus in our interactions with Him, we are interacting with God Himself.