The months of the Jewish year are called in the Torah by number only (the first month, second month, etc.) Over time, during the exile, the months assumed the names given to them by host cultures and thus “Jewish” months as we know them today are actually Babylonian in origin. These names were so common, that 8 out of 12 are mentioned in the later books of the prophets.
Even though the name Av is Babylonian in origin, one cannot help but take note of the subtle nuance of the name. Av means father, and in the fifth month of the Hebrew year, God’s persona of Father is truly demonstrated.
It is stated in the Book of Proverbs (13:24): “One who spares his rod hates his child, but he who loves him, disciplines him in his youth.” God warned the Jewish people that their misguided behavior would result in disaster, but they ignored His warnings. Thus the beginning of the month of Av was the time of the destruction of both Holy Temples, disasters which the Jewish community commemorate with an annual day of mourning on the ninth of Av (Tisha B’Av). When He allowed the Babylonians (and then the Romans) to conquer Jerusalem, destroy the Holy Temple(s) and drive the Jewish people into exile, God had one fatherly goal in mind--that the Jewish people should see the error of their ways and correct themselves.
A parent who punishes a child still loves the child and still wishes to share in the child’s happiness. Rejoicing is also an important facet of the month of Av. Tu B’Av (literally 15th of Av) is a day of tremendous rejoicing in Israel when, traditionally, unmarried maidens would go out to the field to find a husband. Thus in Av, after God completes the role of disciplinarian, He comes forward to watch, and enjoy, as His children rejoice.
Source: Jewish Treats National Jewish Outreach Program
"Chodesh Av," the Month of Av, is referred to in the Bible as the "Fifth Month," counting from Nisan, the "First Month." There is a Biblical reference in BaMidbar (33:38), where we find "And Aharon the Priest went up to Hor Hahar at the command of G-d, and he died there in the fortieth year after the Jewish People had left Egypt, in the fifth month, on the first of the month." (BaMidbar 33:38). This tragedy, the death of Aharon, was perhaps a signal that this month would be a month of tragedy throughout most of Jewish History. See Biblical Significance for an elaboration of this idea.