The Machtzis Hashekel (O.Ch.417) quotes the Hagahos Mordechai in the name of the Tosefta, that there is no Rosh Chodesh Bentching for the month of Av.
Many Poskim argue and hold that we do Bentch on Shabbos Rosh Chodesh Av. The Minhag is to bentch Chodesh Menachem Av, even though the month is Called Av (no Menachem). We say it so that we should be zoche to have the
Geulah etc. (Aruch Hashulchan E.H. 126)
There are sheilos if a Get had the word Menachem in it , if it is kosher.
They once asked from The Tchebiner Rav zt”l how to date letters & documents written in the month of Av.
He told them the answer can be found in the Teshuvos Chasam Sofer, Orach Chaim.
They came back to the Tchebiner Rav and told him they searched the whole Sefer and couldn’t find any such discussion.
He told them , “when learning a Teshuvas Chasam Sofer, it is not enough to learn the text, but one needs to learn the date as well.
If you check Teshuvos # 32, 33, 35, 37, The ones written before Tisha B’av the Chasam Sofer dates it as Av. The ones written after Tisha B’av he dates it Menachem
or Menachem Av.