One does not say a Hesped on Rosh Chodesh, unless the person is a Torah Sage, and his body is present at the time of the Hesped. Some opinions rule that a Hesped should not be said even on the night of Rosh Chodesh.
B. Tziduk Hadin:
One does not say Tziduk Hadin, or the Kaddish that follows it, on Rosh Chodesh. This applies even by the burial of a Torah Sage. This law applies for all days that Tachanun is not said.
May one visit a cemetery on Rosh Chodesh?
Some Poskim write one is not to visit a cemetery on Rosh Chodesh. However today the custom is to allow visiting graves on Rosh Chodesh. Nevertheless, one is not to recite the Maaneh Lashon or other lamentations, although he may ask requests or recite Tehillim. One may certainly visit the grave of a Tzaddik on Rosh Chodesh to pray for an ill person.
May a Matzeiva [tombstone] be established on Rosh Chodesh?
This may not be done if eulogies will take place during the occasion.
 Michaber 420/1 and Y.D. 401/5; Taz 420/1; M”A 420/1; Admur 429/8 regarding the month of Nissan; Chochmas Adam 169/25 [unlike Chayeh Adam 118/7]; Kaf Hachaim 420/1; Nitei Gavriel Aveilus 52/3
 Michaber 420/1 and Y.D. 401/5; Poskim ibid; However see Gilyon Maharsha 344 in name of Dvar Shmuel that a Hesped may be given on Rosh Chodesh Av. See Nitei Gavriel ibid footnote 4
Explanation of source: This law if learned from the law brought in Poskim ibid that women do not lament by funerals during these days. It is also learned from the Poskim in next Halacha who prohibit saying Tziduk Hadin on any day that Tachanun is omitted, and if Tziduk Hadin is omitted than certainly a Hesped may not be said. [See Taz ibid that a Hesped is more severe than Tziduk Hadin as Tziduk Hadin is “not a eulogy but recognition and acceptance of the Divinje decree” and hence some opinions allow Tziduk Hadin; See also M”A 548/8 that one may not even say praise of the dead because this can lead to a eulogy.]
 Michaber Y.D. 401/5; See Admur ibid; Taz and M”A ibid; Chochmas Adam 169/25 [unlike Chayeh Adam 118/7]
 Kol Bo brought in Rama 401/6 regarding Tziduk Hadin, and the same would applt to a Hesped
 Rama 420/2 and Y.D. 401/6; Maharitz Geios; See Kaf Hachaim 420/2
Other opinions: Some Poskim rule that Tziduk Hadin and Kaddish is recited on Rosh Chodesh being it is not considered a eulogy but merely a blessing to Hashem. [Michaber 420/2; Opinions in Tur 420; Talmidei Rashi; Rambam, brought in Taz ibid] Practically, each community is to follow their custom, and in a place where there is no set custom, it is better not to say it. [Kaf Hachaim ibid]
 Taz 420/1; M”A 420/1; See however Tur ibid who brings opinions that say it on behalf of a great Torah Sage.
 Rama ibid
 See Gesher Hachaim chapter 29; Nitei Gavriel 83/1-4
 Kol Bo brought in Beir Hagoleh Yoreh Deah 344/20 [regarding Rosh Chodesh even if falls on seventh of Shiva]; Shaar Hayichudim of Arizal p. 5 [the souls are not by the graves]; Alef Lamagen 581/110; Mishmeres Shalom 5/129
 The reason: As on these days the souls of the Tzaddik are not found in their graves. [Arizal in Shaar Hayichudim p. 5]
 Gesher Hachaim Vol. 1 ch. 29 [p. 259]; Nitei Gavriel 83/3 footnote 3; Daas Torah 581 [brought in Nitei gavriel ibid]
The reason: As the avoidance is only to prostrate oneself on the grave and recite the accustomed lamentations [Maaneh Lashon]. However to pray by the grave was never intended to be forbidden even by the above Poskim. [Gesher Hachaim]
 Nitei Gavriel ibid
 Mishmeres Shalom 5/29
 See Nitei Gavriel Aveilus 2 76/12 p. 576
 Ikarei Hadaat 36/35; Kaf Hachaim 568/94 regarding a Yartzite that falls on Shabbos; See also Likkutei Sichos 19 p. 130 regarding Tzedaka; Igros Kodesh 14/324; Sefer Hasichos 10th Shevat 1957
 Gesher Hachaim 32/5
 See Dvar Yehoshua 1/80; Minchas Yitzchak 3/51-52; Nitei Gavriel 16/3; Piskeiy Teshuvos 697/1