"Look at the love with which she’s put it on him - I think he’d like to know about that - I wish I could tell him."
-- Albus Potter, watching his grandmother Lily (CC4.5)
As a grown-up, a blanket is the only thing that Harry Potter has from his mother. Since he received it after Aunt Petunia’s death from his cousin Dudley Dursley, Harry has always spent time holding the blanket on the date of James and Lily Potter’s murder (CC1.7).
He tries to give the blanket to his son Albus and explain its significance, but they don’t understand each other and it ends in a hurtful quarrel. Albus calls the blanket “mouldy” and throws it away in anger. It then collides with and shatters a bottle of Love Potion (a gift from Ron Weasley, his uncle) (CC1.7).
When later stranded in the past with Scorpius Malfoy in Godric's Hollow (after the Experimental Time Turner is destroyed), Albus realises that he can write a message on the blanket using stolen potion ingredients, a message that should be read by Harry on the right future anniversary date (CC4.5, CC4.6).
Harry doesn't know why his Aunt Petunia kept the blanket (described as "blankets", PS1) hidden away, but he is grateful that she did (CC1.7).
The blanket represents love and the importance of love - Lily Potter's love for her son Harry, Harry Potter's love for his son Albus and his mother Lily, Albus Potter's love for his father Harry and the grandparents he never knew. It connects them and, in the end, love saves them.
From the Web
About the play: http://www.harrypottertheplay.com