While the appearance of the expression "eternal Spirit" is singular (technical term: hapax logomenon), there are numerous contextual clues which might explain why the writer of Hebrews used the expression.
First is the context of the entire book of Hebrews with its theme of the superiority of Christ as contrasted to the inferior--though necessary for a time--Law of Moses, particularly as it applied to the rules and regulations of the sacrificial system of the tabernacle and, later, temple worship.
The writer of Hebrews fleshes out This superiority in numerous ways, but if we limit our focus primarily to Hebrews chapter 9, the author elaborates on that superiority through his use of contrasting terms, particularly present time and eternal.
The writer considers the inherent limitations of the temporal aspects of the "first covenant" or "old tabernacle" (9:1,8), particularly their rules, regulations, and methods. He then contrasts them with the perfections of the "new covenant" (9:15).
Warren Wiersby summarizes the inherent limitations well:
The Old Covenant sanctuary was inferior for five reasons. It was an earthly sanctuary (v. 1), it was a type of something greater (vv. 2-5), and it was inaccessible to the people (vv. 6-7). Furthermore it was only temporary (v. 8) and its ministry was external rather than internal (vv. 9-10) [at NET Bible, Constable's Notes for Hebrews 9:6-10, footnote 280].
On the other hand, the perfections are simply the "opposite" of the limitations
which Wiersbe summarized:
a heavenly sanctuary; the very throne room of heaven, or the heavenly sanctuary, where Jesus presented to his Father, symbolically, the blood of his perfect, once-and-for-all-time sacrifice
the antitype, Christ, who is by definition the fulfillment and embodiment of the types and shadows of the old covenant
a completely accessible way, through faith, and by way of the blood of Jesus
eternal in the heavens (more on that, later)
internal and permanent cleansing within the consciences of repentant sinners (or "the called" of 9:15)
Second is the context of chapter 9, which elucidates the contrast between what is temporal and what is eternal. This contrast, then, seems to be the basis of the expression "eternal Spirit." Whether the word spirit should be capitalized or not is not relevant to this contrast, since either word is perfectly apt!
On the one hand are the "external regulations" (9:1 and 10), the "outer tent" and the "inner tent" (9: 1, 6, 7), and the continual entering of the Levitical priests and the once-a-year entering of the high priest (9:6 and 7). Altogether they comprise the letter of the law.
On the other hand are the eternal verities of which the regulations and modalities of the old covenant with its two priesthoods (Levitical and Aaronic) were but mere shadows. Put differently, they were verities cloaked in mystery and characterized by anticipation of better things to come; that is: Christ, in all his perfections. They comprise the spirit (or Spirit) of the Law.
In conclusion, the phrase "eternal Spirit" is the counterpart of, or antithesis to, all the earthly and temporary regulations of the Law of Moses as touching the Hebrews' cultus. Though the writer to the Hebrews may not have articulated the doctrine of the eternal counsels of God, as Paul did in his writings (e.g., Ephesians 1:11), the writer alludes to it in other, different words, which together give us what is close to being a full-orbed description of a salvation which was planned in eternity but was accomplished on earth in "the fullness of time" (Galatians 4:4).
What preceded the former was the determination of God's eternal counsels to reveal the eternal Spirit of the Word of God in a body of flesh, which despite the Word's self-imposed limitations (the kenosis or self-emptying of Jesus, elaborated in Philippians chapter 2), revealed to true believers the
glory as of the only begotten from the Father, full of grace and truth (John 1:14).