In hebrew, B'nai means Sons. Tzadok means Righteousness. Beit means house (it is also the second letter of the hebrew alephbet) and Shalom means Peace. So...the overall meaning is: Sons of Righteousness, House of Peace.
Why this name? We are the sons of YHVH, and He is the Righteous One. When we enter into His everlasting Kingdom through Yeshua, we are His children, children of Promise. We are the children of Righteousness, not of our own, but that which YHVH sees us in Yeshua.
Ezekiel 44 shows us the lifestyle and fulfillment of who the Son's of Tzadok are, how to conduct their lives and Who they live to serve. We believe that we are called to be kings and priests; as such, we desire to do only that which the Father shows. As sons of Righteousness, we walk in the integrity of Torah, knowing It is the revealed character of YHVH and that we are to be as Yeshua is, walking out the covenant of YHVH, living through Him and dead to our own flesh.
Likewise, Yeshua is our House of Peace. Everywhere we tread, our feet are to leave behind paths of peace which lead us and others to HIm. We will dwell in His House forever; for now, we will enter in as often as possible to glimps His Presence and His true Shalom!
Even as we celebrate Shabbat, we enter into the true Rest of our Father. We long to have that peace always, and we can. If we live in the Kingdom of Heaven here and now, He is ever before us and interceding on our behalf. That is our peace....and so much more.