The Lace Handkerchief is an item in the game Ib. It was given to Ib by her mother on her ninth birthday and has Ib's name inscribed on it. It can only be used when Garry cuts his hand on glass after burning down Mary's painting. The option for it is only available if Ib's bond with Garry is high enough.
When Ib jumps through the Fabricated World painting to return to her own world and meets up with Garry at the Rose Sculpture, he will have forgotten everything that has happened to them; and when he finds the handkerchief in his pocket, he will remember Ib's name, as well as their adventure together. He wil not give back the handkerchief because it is stained with his blood. By keeping the handkerchief and planning to clean it to return it to her at a later time, he promises that they'll meet again.