Although we are looking in context, the question of what “this same Priesthood” refers to in Moses 6:7 is not precisely clear. Even within a scope of just three verses there are several possibilities.
An argument could be made for Priesthood referring to the genealogy mentioned as a book of remembrance. We understand from an overall look at the subject of Priesthood, as the organization of Heaven and the authority of God, that it is always in conjunction with family and is, particularly, synonymous with godly parenthood. In this sense it follows that “this same Priesthood, which was in the beginning, shall be in the end also”, for in the end we will once again have a chain of genealogy linked back to Adam. This view would also be supported by Ezra 2:62 and Nehemiah 7:64, in which absence of a genealogy barred certain peoples from the Priesthood.
However, verse 8 seems to discuss genealogy as a separate, albeit related, topic. This idea of Priesthood as parenthood may also lead us to another idea. A certain emphasis is placed, in verse 5, on the language of Adam, in which was kept the book of remembrance and which was taught to the children of Adam. In the scriptures, particularly in the modern scripture given by Joseph Smith, we see language frequently used broadly to mean the overarching subject of discussion. In this sense it doesn’t mean an entirely distinct phonology or syntax, but a special topic that probably has some amount of its own vocabulary. Such is the idea used when Nephi refers to Lehi’s rejoicings or rebukings and says, “After this manner was the language…” (1 Ne. 1:15; 3:21; 5:3). So too Priesthood could be referring to that special language of the prophets that contains such concepts as repentance, sanctification, salvation, and even genealogy. This parent-taught language, which included a writing system and may or may not have been possessed by the lines of Cain and his followers, would certainly have held as its topics sacred subjects not discussed by those who apostatized from the Children of God.
This idea of Priesthood as the language of the first parents – which, we know, will indeed be restored at the last days (Zeph. 3:9) – could also lead us to an even more precise concept of what “this same Priesthood” is: it might, in particular, be the revelations that were had by the fathers, kept in their language, and handed down by parents in the book of remembrance. Essentially, this would make it the scriptures and the words of God, given by the Spirit. Such a view is supported by the knowledge that by the Priesthood the worlds were created (see Gospel Principles, ch. 13, Par. 2, 2009) and also “by the Word” was the creation wrought (1 John 1; 2 Pet. 3:5) – the Priesthood, in its most sublime sense, is in general the language and in particular the word of The Father, of which Christ is the embodiment, conduit, and fulfilling agent.