The Mezuzah on the Right and the Hanukkah Menorah on the Left
The Talmud in Tractate Shabbat states that one should place the Hanukkah Menorah to the left of the door and the Mezuzah to the right. This is referring to the original custom where people would light their Menorahs outside their front door to the left of the door. They would place the Menorah very low to the ground in order to make it perfectly clear that the lights were not for our use, but only to publicize the Hanukkah miracle. By having the Mezuzah on one side and the Menorah on the other side, you would be surrounded by Mitzvot (good deeds) when you entered your home.
Later during medieval times it became unsafe to light the Menorah outside the house where all could see it. It became customary in most places to light the Menorah inside the home. At least the miracle would be publicized in this way to one's own family and friends, if not to the world at large.
Today in Jerusalem and many other places in Israel, it is still customary to light the Menorah on the outside to the left of the door, opposite the Mezuzah.
The Dedication of the Temple
There is another connection between the Hanukkah Menorah and the Mezuzah. It is the idea of dedication. The Menorah commemorates the re-dedication of the Temple in Jerusalem after it had been defiled and turned into a pagan temple by the Seleucids (Greek speaking Syrians). The oil for the Menorah of the Temple should have lasted only one day but it lasted eight allowing for the production of new ritually pure oil. The eight candles of our Menorah commemorate this re-dedication miracle. In fact the word "Hanukkah" means "Dedication" in Hebrew.
Now that we have no Temple, we are taught in the Talmud that God dwells in the synagogues and in the homes of those who dedicate their home to God. This is one of the reasons why we put Mezuzahs on each and every door of our home. By placing a Mezuzah on each room, we show that our home is dedicated to the service of God.
A Mezuzah - The Perfect Hanukkah Gift!
Giving gifts on Hanukkah is not an original Jewish custom. It is something that has developed in America mostly as a Jewish response to Christmas presents. American Jews did not want their children to feel left out of the gift receiving of the holiday season and thereby be attracted to other religions.
But the gift of a Mezuzah is actually in keeping with the true spirit of Hanukkah which is all about rededicating our lives to service of God. What better gift can you give for Hanukkah than one which so perfectly expresses the spirit of the holiday?
Mezuzahs are also a great Hanukkah gift since they come in so many styles and colors that allow you to buy something that fits the taste of the recipient. Also since many people don't have Mezuzahs on every door of their home, they can always use another one.