Is there a special word or phrase for a CD with a carefully picked playlist you made for a particular person, often given as a gift?
It is called a mix CD. (From mixtape, originally)
As an alternative to 'mix CD,' I've often just called them compilations.
The only problem with calling them simply 'compilations' is that it doesn't really differentiate from a commercially released compilation album, so you would have to say 'my own compilation' or 'compilation that I made.'
So it's not nearly as nice and succinct as simply 'mix CD,' but it still works.