From BiblePortal Wikipedia

Smith's Bible Dictionary [1]

Beth-a'ven. (House Of Nothingness, that is, Of Idols). A place on the mountains of Benjamin, east of Bethel ,  Joshua 7:2;  Joshua 18:12, and lying between that place and Michmash .  1 Samuel 13:5;  1 Samuel 14:28.

In  Hosea 4:15;  Hosea 5:8;  Hosea 10:5, the name is transferred to the neighboring Bethel , - once the "House Of God", but then the "House Of Idols" or "Naught".

Fausset's Bible Dictionary [2]

("house of nothingness or vanity".) On the mountains of Benjamin, E. of Bethel ( Joshua 7:2;  Joshua 18:12), between it and Michmash  1 Samuel 13:5;  1 Samuel 14:23). Near it was the "wilderness," i.e. pasture land of Bethaven ( Joshua 18:12.) In  Hosea 4:15;  Hosea 5:1;  Hosea 10:5 Bethel, "house of God," is called Bethaven, "house of vanity", because of Jeroboam's golden calf.

Morrish Bible Dictionary [3]

A place or 'wilderness' of Benjamin near Bethel.  Joshua 7:2;  Joshua 18:12;  1 Samuel 13:5;  1 Samuel 14:23;  Hosea 4 .  15;  Hosea 5:8;  Hosea 10:5 . Though this is said to be on the east of Bethel, in Hosea it would appear to be a name given to Bethel itself as being no longer the 'house of God,' but the 'house of vanity' because of the idols there.

Watson's Biblical & Theological Dictionary [4]

the same with Bethel. This city, upon the revolt of the ten tribes, belonged to the kingdom of Israel, and was therefore one of the cities in which Jeroboam set up his golden calves. Whence the prophet in derision calls it, "Bethaven," the house of vanity or idols,   Hosea 4:15 , instead of "Bethel," the house of God, the name which Jacob formerly gave it, when he had the vision of the mysterious ladder, reaching from earth to heaven,   Genesis 28:19 .

Kitto's Popular Cyclopedia of Biblial Literature [5]

Beth–A´ven (house of idols), a nickname for the town of Bethel, applied to it after it became the seat of the worship of the golden calves [BETHEL]. There was, however, a town of this name not far from Bethel eastward ( Joshua 7:2;  1 Samuel 13:5), the existence of which, perhaps, occasioned the transfer of the name to Bethel. There was also a desert of the same name ( Joshua 18:12).