Based on On Tish'a B'Av,can someone say the typical "Mourner's statement" to any/everyone else?, may one perform the mitzvah of Nichum Aveilim for one who is sitting shiv'a on Tish'a B'av? Would it make a difference as to when during the day it was, or what the person was saying to console/comfort the mourner?
Yes, one is allowed to be מנחם אבלים at any time on Tisha B'av. Rabbi Nosson Gestetner explains, that from the fact that the tefillah of Nacheim is said in Shemone Esrei on Tisha B'av, we can derive that there is no Issur to accept condolences- ניחום on Tisha B'av. And even though we only say it in Shemone Esrei of Mincha (למנהג הרמ"א), someone who doesn't fast and has to Bentch Birkas Hamazon will say Nacheim then even if he eats at night (משמעות הרמ"א סימן תקנ"ז). This proves that at all times on Tisha B'av is one allowed to accept condolences. שו"ת להורות נתן חלק י' (סימן נב אות ד)
Also see ספר נחמת ישראל (סוף פרק לו) בשם הגר"ח קניבסקי זצוק"ל