We don't light a menorah

Growing up as an Orthodox Jew in New York, a rabbi's son no less, I thought I knew all there was to know about Hanukkah; but soon after arriving at yeshiva in Israel at age 17, I discovered that I had it all wrong. See, gelt was really demei Hanukka, and it wasn't so much chocolate coins as gifts. Also, that's not a dreidel, it's a sevivon, and the letters are wrong. Plus, they're not latkes, they're levivot, and no one really eats them because they've already gorged themselves on sufganiyot, so-called because they absorb all moisture in your stomach and swell like a sponge (sefog). [Your etymology may vary.]
Still, most shocking was the fact that the thing we light with all the branches is not a menorah, but a hanukkiah. I was confused because, unlike latkes, dreidel and gelt, menorah was already a Hebrew word. But in Israel, menorot were to be found in the lighting department of the local hardware store, while The Menorah was that seven-branched vessel of the Temple, known to the world from the Arch of Titus, featured in the Emblem of the State of Israel. A menorah is simply a lamp, but if it is that special nine-branched version for Hanukkah, eight for the eight nights plus one for the shamash, which kindles all the others, it becomes a hanukkiah.
So where did this neologism come from? Lexicographer Ben-Yehuda, but not the one you're thinking of: Hemda popularized the term in 1897. Nor was she the first. In fact, tonight, the fifth night of Hanukkah, marks the 249th yahrtzeit of Jerusalem-born Rabbi Abraham Meyuhas, who writes (Sedeh Haaretz III OH 38) about "a brass candlestick which we call a hanukkiah, to which the artisan affixed an additional light which we call a shamash, to be used for its illumination instead of the others, which are the essence of the mitzvah, as we are forbidden to use them for illumination." Rabbi Meyuhas knew what a hanukkiah was, though he was afraid his readers might not.
It is significant that the menorah used for Hanukkah has a special name. The fact is that there is no halakhic significance per se to the vessel; you could stick one wax candle on your windowsill and fulfill the mitzva. It was only in the medieval era that Jews started crafting particular vessels to be used only for Hanukkah; indeed, for a religion which strictly forbids graven images, this was an opportunity for artistic expression, like the wine-cup used at Sabbath's onset and the spice-box used as its conclusion.
And thus we come to the paradox at the heart of Hanukkah. On the one hand, it is a celebration of the Temple; on the other hand, it marks the ascension of the dynasty which would ultimately welcome the Roman Empire, demolishers of the Temple, into Jerusalem.
While the biblical Menorah -- that famous seven-branched one -- may be be physically more impressive, it was lit for less than 1,500 years according to the traditional counts of all the various incarnations of the Tabernacles and the Temples. Meanwhile, the lights of Hanukkah have been kindled for nearly 2,200 years, uninterrupted.
In fact, according to Jewish lore (Midrash Tanhuma, Behaalotekha 3; Talmud Menahot 29a), the Menorah was not actually made by humans at all. Moses was so perplexed by its intricate details that God told him to throw it in the fire, and out popped the Menorah fully-formed. The hanukkiah, on the other hand, is a wholly human invention.
Even in the recounting of the miracle in our daily prayers throughout Hanukkah, we do not refer at all to the Menorah inside the Temple. Instead, we say "they kindled lights in Your holy courtyards." The Menorah and the hanukkiah aren't the same.
Which brings us back to the emblem of the State of Israel. It definitely features a seven-branched candelabrum in the center--but this is flanked by two olive branches. That's not an aesthetic flourish, but the vision of Zechariah (ch. 4), whose words we read on the Sabbath of Hanukkah. The two olive branches represent the religious and secular leadership of the people, and together with the seven bronze branches, they make what we could call a hanukkiah. (I leave it up to the reader's discretion to decide which one is the shamash.)
This is the beauty of the State of Israel. It does not spring fully-formed from the fire. It does not descend from Heaven. It is made by flawed human beings, a combination of natural growth and technological artistry. It is a construct of the spirit. And we are charged by our faith to constantly refashion it into a more perfect union.