Awoken Mementos are a type of cosmetic item in Destiny 2 that offer a specific unique shader, a weapon title, and a tracker aligned with the source of the Memento. They can only be used on crafted weapons, which must reach level milestones before certain Memento features are available. There are currently three types of Mementos available in Destiny 2: the Gambit Memento, Nightfall Memento, and Trials Memento. The steps for obtaining each Memento can vary, but players should keep in mind that they can only hold one of each type of Memento.
If you have obtained the desired Memento, you can visit The Enclave on Savathûns Throne World to approach the Relic to either reshape an existing weapon or craft a new one if needed. In the crafting menu, the Memento slot is between the perks and the "Finalize Shape" option. There is no cost for this process other than the Memento itself.
It is worth noting that there is a specific quest called Hunters Remembrance that requires players to purify three Awoken Mementos by hunting down three enemies. After purifying the Mementos, players can turn them in to an Awoken statue in exchange for the Wish-Ender Exotic Bow.