The Gift Menu should look something like the below image:
Though your screen may differ, especially if you don't have any gifts! If you have purchased a morale gift from the store, it will be displayed in a list on the left side of the screen. On the right side of the screen you will see a list of players and their current morale. Select the Give Item button (or press the Enter key) to give the selected item to the selected player and improve their morale! The Back button, like most other Back buttons, will return you to the Town Menu.
One item of note, certain players like some gifts more than others.