Timelost Weapon Bounties are new Bounties that have come to Destiny 2. You can purchase them at an Obelisk for 10 Polarized Fractaline and 3000 Glimmer. To awaken the Obelisks, you will need to play through the Matter of Time and Keeping Time quests. You can earn Polarized Fractaline by completing quests, Triumphs, and runs in the Sundial.
You have to find these weapons in time because they were lost, and you need to perform an incredible feat to retrieve them.
You cannot purchase Timelost Weapon Bounties from Obelisks straight away. You need to increase their Resonance Rank first by doing different Timelost Weapon Bounties each week. The weapons become available at Rank 1, Rank 3, and Rank 5. Ranking the Obelisks requires Polarized Fractaline, so this will be an essential currency this season.
Each bounty gives you a random objective to complete, such as killing a certain amount of enemies with your Super or killing a certain number of Guardians in the Crucible. Upon completion of the bounty, you can hand it in from your Quest screen, and the weapon reward goes to your inventory. You can only complete each Bounty more than once, but they will have limited availability until you can rank up the Obelisks.
You will need to complete three of these Bounties as one of the steps in the Keeping Time Quest.