Let’s consider each:
The concept of getting a name from God was predicted in OT passages like Isaiah 62:2 and 65:15. Some commentators believe that the name on the stone is the name of God (Rev. 3:12), which Christ also has (Rev. 19:12). However, this doesn’t seem to fit with the notion that only the believer would know this name. In the OT, the concept of a name was closely associated with one’s identity. This concept of a name becomes important at the end of the book, when only those names found in the book of life will enter heaven (Rev. 20:15).
The historical backdrop of Pergamum gives us some insight into the symbolism of the white stone:
First, the white stone imagery comes from trial procedures in Pergamum. Osborne writes, “In ancient trials jurors would cast a white or black stone into an urn to vote for acquittal or guilt (cf. Acts 26:10); while no name was written on the stones, the trial setting could make sense in the Pergamene situation.” Jesus could be communicating their acquittal from guilt. Walvoord writes, “In courts of law being given a white stone is thought to represent acquittal in contrast to a black stone which would indicate condemnation.”
Second, the white stone imagery comes from the Pergamum games. Osborne writes, “It was common for members of a guild or victors at the games to use stones as a ticket for admission to feasts, and also for free food or entrance to the games.” Thus Jesus could be explaining that because of the white stone, these believers get free access into the feast of God: heaven.
 Osborne, Grant. Revelation. Baker Exegetical Commentary on the New Testament. Grand Rapids, MI: Baker Academic. 2002. 138-139.
 Walvoord, John. The Revelation of Jesus Christ. JFW Publishing Trust. Chicago, IL. 1966. 70.
 Osborne, Grant. Revelation. Baker Exegetical Commentary on the New Testament. Grand Rapids, MI: Baker Academic. 2002. 139.