Here is some useful information regarding this episode:
Needless difficulty has been felt in explaining this incident in consequence of a somewhat defective translation. The path of the prophet Elisha lay through the district of Bethel, the stronghold of idolatry in Israel (1 Kings xii. 28—33), where, as in Dan, stood one of the golden calves set up by Jeroboam.
In this place insult offered to Jehovah's prophet would be intended as insult to Jehovah, and, so regarded, it was properly met by an immediate and terrible punishment.
It appears that there was a number of idle young men on the outskirts of the town, lawless, rude, and amusing themselves with rough play. They are called " children," but the same Hebrew word is used in 1 Kings xii. 8, 10, 14, where it is applied to young men of the same age as King Rehoboam. In all the languages of the East the words "child" and "children" often denote simply a social relation, and are constantly applied to full-grown persons, as in the New Testament.
"No one who has travelled in the East can have failed to notice the extreme lawlessness of a certain class of boys and young men living on the outskirts of a town, especially toward a Jew, a Christian, or a European, who should happen to be passing by alone or unprotected. Let him go, for instance, to the castle hill of Smyrna, and, if it be a holiday and the 'boys' (oghlans) are out, he will perceive stones whizzing past him, and will hear the shouts of ' Frank,' ' hat-wearer'-- rallying the rowdies of the vicinity, and warning him to beat a hasty retreat."