According to author Randy Alcorn, in his book Heaven, we surely will. Alcorn starts building his case from the contents of Rev 6.
As we’ve seen, the martyrs depicted in Revelation 6 clearly remember at least some of what happened on Earth, including that they underwent great suffering. If they remember their martyrdom, there’s no reason to assume they would forget other aspects of their earthly lives. In fact, we’ll all likely remember much more in Heaven than we do on Earth, and we will probably be able to see how God and angels intervened on our behalf when we didn’t realize it.
Alcorn points out the fact that we are promised, in Luke 16:25, to be comforted in Heaven about the bad things that happened to us on earth. That comfort implies that we will remember what it is we are being being comforted for.
Additionally, there is the fact that we will give an account of our earthly lives after we die (see 2 Corinthians 5:10; Matthew 12:36). Alcorn argues that
[g]iven our improved minds and clear thinking, our memory should be more— not less—acute concerning our life on Earth . Certainly, we must remember the things we’ll give an account for. Because we’ll be held accountable for more than we presently remember, presumably our memory will be far better.
But perhaps his strongest argument for memory of our earthly lives is the doctrine of eternal rewards. Alcorn explains:
The doctrine of eternal rewards hinges on specific acts of faithfulness done on Earth that survive the believer’s judgment and are brought into Heaven with us (1 Corinthians 3:14). In Heaven, the Bride’s wedding dress stands for “the righteous acts of the saints” done on Earth (Revelation 19:7-8). Our righteous deeds on Earth will not be forgotten but “will follow” us to Heaven (Revelation 14:13).
The positions of authority and the treasures we’re granted in Heaven will perpetually remind us of our life on Earth, because what we do on Earth will earn us those rewards (Matthew 6: 19-21; 19:21; Luke 12: 33; 19: 17, 19; 1 Timothy 6:19; Revelation 2:26-28). God keeps a record in Heaven of what people do on Earth, both unbelievers and believers. We know that record will outlast our life on Earth— for believers, at least until the judgment seat of Christ (2 Corinthians 5: 10); for unbelievers, right up until the Great White Throne Judgment (Revelation 20:11-13), just preceding the coming of the new heavens and New Earth. For those now in Heaven, these records of life on Earth still exist.
Alcorn concludes by reminding us of the importance of our memories to our personalities:
Memory is a basic element of personality. If we are truly ourselves in Heaven, there must be continuity of memory from Earth to Heaven . We will not be different people, but the same people marvelously relocated and transformed. Heaven cleanses us but does not revise or extinguish our origins or history. Undoubtedly we will remember God’s works of grace in our lives that comforted, assured, sustained, and empowered us to live for him.