Part with
Definition 1 of 1
Pro Tip 1/3Emotional Value
The phrase implies an emotional attachment to the thing being given away. 
I finally decided to part with my childhood toys.
Pro Tip 2/3Reluctance Express
'Part with' suggests the act is done unwillingly or with some hesitation. 
She had to part with her cherished paintings to cover the debt.
Pro Tip 3/3Not Just Selling
It can also mean giving away without any financial transaction. 
He couldn't part with the letters from his pen pal.