The earliest 2 sources for Chanukah are the talmud in Shabbos 21b and the Al Hanisim prayer. Both of these mention Praise and Thanksgiving, but neither mentions happiness per se.
However, it seems clear for another reason that Chanukah is associated with happiness specifically. We recite Hallel, which is an expression of joy. Rambam writes in משנה תורה, הלכות מגילה וחנוכה in law 3:6 that:
וְלֹא הַלֵּל שֶׁל חֲנֻכָּה בִּלְבַד הוּא שֶׁמִּדִּבְרֵי סוֹפְרִים אֶלָּא קְרִיאַת הַהַלֵּל לְעוֹלָם מִדִּבְרֵי סוֹפְרִים בְּכָל הַיָּמִים שֶׁגּוֹמְרִין בָּהֶן אֶת הַהַלֵּל. וּשְׁמוֹנָה עָשָׂר יוֹם בַּשָּׁנָה מִצְוָה לִגְמֹר בָּהֶן אֶת הַהַלֵּל. וְאִלּוּ הֵן. שְׁמוֹנַת יְמֵי הֶחָג. וּשְׁמוֹנַת יְמֵי חֲנֻכָּה. וְרִאשׁוֹן שֶׁל פֶּסַח וְיוֹם עֲצֶרֶת. אֲבָל רֹאשׁ הַשָּׁנָה וְיוֹם הַכִּפּוּרִים אֵין בָּהֶן הַלֵּל לְפִי שֶׁהֵן יְמֵי תְּשׁוּבָה וְיִרְאָה וָפַחַד לֹא יְמֵי שִׂמְחָה יְתֵרָה. וְלֹא תִּקְּנוּ הַלֵּל בְּפוּרִים שֶׁקְּרִיאַת הַמְּגִלָּה הִיא הַהַלֵּל:
And it is not only the Hallel of Chanukah that is from the words of
the Scribes. Rather the reading of the Hallel is always from the words
of the Scribes on all of the days in which we complete the Hallel. And
it is a commandment to complete the Hallel on eighteen days in the
year. And these are them: The eight days of the Festival (Sukkot);
eight days of Chanukah; the first day of Passover; and the day of
[Shavuot]. But there is no Hallel on Rosh Hashanah and Yom Kippur
because they are days of repentance, awe and fear - not days of
excessive joy. And they did not ordain Hallel on Purim, because the
Scroll [of Esther that is read on it] is the Hallel.
The clear implication is that all days of Hallel are indeed days of excessive joy. Thus happy chanukah is a highly appropriate greeting.
Your friends idea is from shulchan aruch who writes in OC 670:2:
ריבוי הסעודות שמרבים בהם הם סעודות הרשות שלא קבעום למשתה ושמחה
and Mishnah Berurah comments:
(ו) שלא קבעום למשתה ושמחה - אלא להלל ולהודות.
but it seems that this statement is meant relative to purim and other festive days, but not that Chanukah is not a happy time at all.
Well, here in the US, who would be wishing each other a "Happy Hanukka" if it weren't a "Merry Xmas" stand-in?
A lot of people, in my opinion. I typically wish people "Happy Purim", "Happy Tu Bishvat", etc. during the appropriate times of year.