Do we say Hallel on Rosh Chodesh Av?
Yes, we do. But the question is not a silly one. The festive flavor usually associated with Rosh Chodesh is hardly in evidence.
The mishna says: "As Av enters, we diminish joy". Rosh Chodesh Av is the beginning of the stricter mourning period for the destruction of the Beit HaMikdash - First and Second. The restrictions of the Nine Days generally apply to Rosh Chodesh Av. (This is the Ashkenazi practice. S'faradim begin the restrictions on the second of the month.) It is one of two months on whose Rosh Chodesh fasting is not forbidden.
Rosh Chodesh Av is the Yahrzeit of Aharon HaKohen. It is the only Yahrzeit mentioned in the Torah. It is recorded, not in Parshat Chukat where we read of Aharon's passing, but in Mas'ei - which we read on the Shabbat closest to Rosh Chodesh Av.
Rosh Chodesh is Rosh Chodesh. It is a joyous and hopeful commemoration of the Beit HaMikdash, not only its destruction. Notwithstanding the mournful nature of the first third of Av, we must keep in mind that after the 10th of the month, the consolation and promise for a brighter future takes over.
Rosh Chodesh Av conjures up a mixture of conflicting moods. That's okay. Is it contradictory to say Hallel on the day that ushers in a sad period? No. Being Jewish often means being able to see the bright side of sad times, and not forget the sad note at happy times.
May we soon merit the coming of Moshiach and the rebuilding of the Beit HaMikdash - and everything that goes with it - and may our joy then become untarnished and complete.
Let us each put into action the qualities of Aharon HaKohen - love peace and pursue it, love people and bring them closer to Torah - so that the times we yearn for will become a reality, speedily in our time, Amen.
Our Thanks to Phil Chernofsky of the OU/NCSY Israel Center for Including This Material in His Remarkable Torah Tidbits