Is one obligated to eat a special meal on Rosh Hodesh?
The Shulhan Aruch (Orah Haim 419) addresses this question and writes explicitly that it is a "Misva" to "increase one’s meal" on Rosh Hodesh. This means that although it is not strictly obligatory to eat special foods on Rosh Hodesh, it is proper to do so, by eating something special that one does not eat on an ordinary weekday. A person does not necessarily have to eat a meal with bread, as is required on Shabbat and Yom Tob, but one should, as mentioned, eat something that he does not eat on normal weekdays, in order to give honor to the occasion of Rosh Hodesh.
The source of this Halacha, as the Kaf Ha’haim (Rav Yosef Haim Sofer, Baghdad-Israel, 1870-1939) writes, is the verse in the Book of Bamidbar (10:10) which states, "U’be’yom Simhatchem U’be’mo’adechem U’b’rosheh Hodshechem" ("On the day of your joy, and on your festivals and on the first of your month"). The Torah here equates Rosh Hodesh with Shabbat and festivals, and it is therefore proper to observe the occasion of Rosh Hodesh as a quasi festival, by eating a special meal.
The Gemara comments that although God determines a person’s livelihood for the entire year on Rosh Hashanah, this decision does not take into account a person’s expenses for Shabbat and other special occasions. The money that a person spends for special foods on these occasions is taken out of God’s account, so-to-speak, and not from the money prescribed for him. Therefore, we should not hesitate to eat special foods on Rosh Hodesh, despite the costs entailed.
The Kaf Ha’haim writes that a "Yereh Shamayim" ("God-fearing person") should make a point of eating bread on Rosh Hodesh and reciting the Beracha over "Lehem Mishne" (two whole loaves), as we do on Shabbat and Yom Tob. And the Hid"a (Rav Haim Yosef David Azulai, 1724-1806) comments that on Rosh Hodesh we experience a "He’arat Ha’nefesh" ("illumination of the soul"), parallel to the extra soul that we receive each week on Shabbat.
Finally, the Kaf Ha’haim and others mention that it is proper for one to wear a special garment on Rosh Hodesh, such as a special shirt or tie. The Vilna Gaon (Rabbi Eliyahu of Vilna, 1720-1797), for example, used to wear his special Shabbat hat on Rosh Hodesh. Some suggested an allusion to this practice in the verse in the Book of Bereshit (45:22) that mentions the five special garments that Yosef gave his brother, Binyamin – "Ve’hamesh Halifot Semalot" ("five changes of clothing"). The word "Hamesh" can be viewed as acronym representing the words "Hodesh," "Mo’ed," "Shabbat," thus alluding to the occasions of Rosh Hodesh, Yom Tob and Shabbat. And on these three occasions, it is proper to wear "Halifot Semalot" – different clothing than we normally wear.
Summary: It is proper to eat something special on Rosh Hodesh in honor of the occasion, a food or meal that one does not normally eat on an ordinary weekday. One should also wear a special garment in honor of Rosh Hodesh, such as a special shirt, tie or hat.