The Tur (Rabbenu Yaakob Ben Asher, Germany-Spain, 1269-1343), in Orah Haim (419), draws three proofs to the fact that one is obligated to have a special Se’uda (meal) on Rosh Hodesh. His first proof is from the comment of the Talmud Yerushalmi (Megilla) that one is allowed to "delay the Rosh Hodesh meal." The Korban Ha’eda commentary to the Yerushalmi (by Rav David Frankel, Germany, 1704-1762) explains this remark to mean that if Rosh Hodesh falls on Shabbat, such that one eats special meals in honor of Shabbat, and there is no indication that one eats a meal in honor of Rosh Hodesh, he delays the Rosh Hodesh meal until Sunday. Somewhat similarly, the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) writes that the Ya’abetz would add a special portion of food to the Melaveh Malka meal on Mosa’eh Shabbat when Rosh Hodesh fell on Shabbat, in honor of Rosh Hodesh. The Ben Ish Hai writes that this is a proper practice to follow. In any event, the Yerushalmi clearly works off the assumption that there is a requirement to eat a special Se’uda on Rosh Hodesh.
The second proof comes from the story told in the Book of Shemuel I (20) which is read as the Haftara on Shabbat Ereb Rosh Hodesh. We read that King Shaul was hostile to David, and so David decided that he would not be present at Shaul’s table on Rosh Hodesh. He asked his friend, Yehonatan – Shaul’s son – to tell Shaul that David could not be present because he had a family feast that day. The Tur notes that this appears to indicate that it was customary for families to have special feasts in honor of Rosh Hodesh. (The Bet Yosef clarifies that if it were not customary for families to have a special Rosh Hodesh meal, then this would not have been a plausible excuse for David’s absence from the royal palace on Rosh Hodesh.)
The Tur’s third proof is a verse in the Book of Bamidbar (10:10) which requires blowing the Hasoserot (trumpets) "Be’yom Simhatchem U’be’mo’adechem U’be’rosheh Hodshechem" – "on your days of joy, on your festivals, and on your Rosh Hodesh days…" This verse clearly links Rosh Hodesh with the Yamim Tobim, suggesting that just like there is an obligation to have a special meal in honor of Yom Tob, one is similarly obligated to have a special meal in honor of Rosh Hodesh.
Interestingly enough, the Shulhan Aruch sets aside an entire Siman (chapter) – Siman 419 – to tell us this brief Halacha, that one is required to have a special meal on Rosh Hodesh. The Shulhan Aruch similarly makes a special Siman earlier – 300 – to teach us the obligation to eat a Melaveh Malka meal on Mosa’eh Shabbat. It seems that these two meals were commonly neglected, and so the Shulhan Aruch found it necessary to make a special Siman for each Halacha, in order to impress upon us the importance of having a meal on Mosa’eh Shabbat and a meal on Rosh Hodesh. (When Eliyahu confronted the prophets of the idol Ba’al at Mount Carmel, he turned to the people and asked, "Ad Matai Atem Posehim Al Sheteh Ha’se’ifim" – "Until when will you be straddling both sides of the fence?" (Melachim I 18:21). One Rabbi said that this could also be read as, "Until when will you skip the two Se’ifim" – referring to the two Se’ifim (passages) in the Shulhan Aruch that talk about these two meals, Melaveh Malka and the Rosh Hodesh Se’uda…)
The Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933) writes that one is not required to eat bread with his Rosh Hodesh meal, whereas the Kaf Ha’haim (Rav Yaakob Haim Sofer, Baghdad-Israel, 1870-1939) writes that one must eat bread, and one should even use two loaves as "Lehem Mishneh," like on Shabbat.
The Pesikta comments that on Rosh Hashanah, G-d determines a person’s livelihood "from Tishri until Tishri" – for the entire coming year – except for "Tishri," meaning, except for "Torah," "Shabbat," "Rosh Hodesh" and "Yamim Tobim." That is to say, the money we spend for Torah education, and for our meals on Shabbat, Yom Tob and Rosh Hodesh, are on "G-d bill," so-to-speak. These funds are not taken from the amount we are decreed to have, and we are guaranteed to be fully reimbursed for whatever we spend to fulfill these Misvot.
Summary: There is a requirement to eat a special meal on Rosh Hodesh. According to some views, one is not required to eat bread with the Rosh Hodesh meal, whereas other require having two loaves of bread, like on Shabbat. Some Poskim ruled that when Rosh Hodesh falls on Shabbat, it is proper to add a portion of food to the Melaveh Malka meal on Mosa’eh Shabbat in honor of Rosh Hodesh.