Its ironic that the concept we should learn from Chanukah is that we are different from the non Jews and how they forced us to give up on our Mitzvos and many Jews willingly gave up and made concessions on their yiddishkiet until they became Greeks (misyavnim) and yet what do we do? We go and give Chanukah gifts which is a Christmas concept stolen from the non Jews.
I think the greatest gift you can give each other is to learn what Chanukah is about and to both show appreciation to Hashem that we live in a society where theres religious freedom and we are able to do the Mitzvos freely and build beautiful Jewish families.
The gift giving you can leave for before or after Chanukah.