Taking my Ph.D. advisor out for dinner after graduation
What is the professional/moral way of doing this?


I think that if you have never invited/been invited to a casual gathering involving the advisor's family, it will be awkward to do so now. If you feel like a one-on-one dinner invitation is too much, how about a lunch appointment? 

I will say this - many universities have strict rules on receiving expensive gifts from students/subordinates. If your meaning of precious is simply "of significant emotional value" (say, you made a piece of art yourself, or found a book that they really like) that's fine; if you mean expensive (say, an expensive watch) - your advisor may not be able to accept the gift at all (or, as is the case in my university, will have to report the gift and then give it to the school).