Hedwig will leave on an adventure then returns. Hedwig is able to carry Flowers, Seeds and Watering Cans in her suitcase. Depending on what you send her with and the order you put the items in, she can bring back rewards and/or Letters.
The Fortuneteller can make Hedwig come back sooner in exchange for watching an ad.
Hedwig can wear the following Mementos:
|+15||Serve Penniless Toad 10 times.|
|+15||Serve Penguin 30 times.|
Question Mark Hat
|+105||Obtained via Gachapon.|
Hedwig is very probably an homage to the eponymous character from the Harry Potter franchise where she is also a female white owl able to deliver messages.