I often see the following from Yerushalmi Berachot cited as a source that Moshiach will be born on Tisha B'Av:
עובדא הוה בחד יהודאי דהוה קאים רדי געת תורתיה קומוי עבר חד ערביי ושמע קלה א"ל בר יודאי בר יודאי שרי תורך ושרי קנקנך דהא חריב בית מוקדשא געת זמן תניינות א"ל בר יודאי בר יודאי קטור תוריך וקטור קנקניך דהא יליד מלכא משיחא
What is the nature of the connection between Tisha B'Av and Moshiach? Are there other sources that discuss this relationship?
It's brought down in some Sichos from the Lubavitcher Rebbe that there's a question: How can Hashem destroy the Beis Hamikdosh? There is halochos against destroying even a Shul. The answer is that it was destroyed in order to build a nicer and better one. So the destruction -tisha b'av- is connected with the rebuilding -moshiach-.
There are several. Some of them are brought here by the Lubavitcher Rebbe.
Kabbalistically Kiddush Levana is said after Tisha B'Av because the afternoon is the time of the birth of Moshiach.
Similarly, Rabbi Chaim Vital (the main student of the Arizal) says we can say Pesukei Nechama - verses of comfort, which are normally skipped on Tisha B'Av, at Mincha because it is the birth time of Moshiach. This brought in Halacha in the Birchei Yosef.
The Pesikta Rabbati says that "Joy will only come on Tisha B'Av" - emphasizing the turnaround from its current status of mourning - which can be connected to the same idea.
Based on the above he says that this is the inner reason why Tisha B'Av is called "moed" - a holiday.
