Combining Matthew Henry's insight here, Constable's insight here, and perhaps a soupçon of my own insight, I suggest we are safe in saying the following:
Elijah's idiomatic expression, and possibly a common Hebraism in Elijah's day, "Go back again, for what have I done to you?" could be paraphrased as follows:
"Go ahead. You're free to do as you like."
Notice that in context, Elisha is obviously a member of the landed gentry. He has eleven servants/employees in front of him plowing the field, each with a pair of oxen, as he plows right along with them. (Perhaps the 12 pair of oxen are symbolic of Elisha's yet future role of providing prophetic leadership to the 12 tribes of Israel and Judah.)
Once Elijah had done as the LORD had commanded him to do; namely, anoint Elisha the son of Shaphat as his acolyte and eventual successor, Elijah's task was complete. By throwing his mantle onto Elisha (see 19:16b and 19:19b), the message was clear to Elisha: he was to be the senior prophet's successor.
In other words, Elijah's job was not to force Elisha to follow him at once. No, Elisha's desire to follow Elijah would have to come from both Elisha and the LORD.
In modern parlance, Elijah was saying to Elisha,
"Hey, dude, right now I'm not the boss of you. If out of respect for your parents you want to say goodbye to them and to your peeps first, go ahead. I've done my job as the LORD commanded me; the rest is a matter between you and the LORD."
Perhaps you, like me, are reminded of Jesus' call to those would-be followers who made excuses for not following him right away. One said,
"'Lord, permit me first to go and bury my father'" (Luke 9:59).
"'I will follow You, Lord; but first permit me to say good-bye to those at home'" (v.61).
Interestingly, notice how Jesus responded to the latter person,
"'No one, after putting his hand to the plow and looking back, is fit for the kingdom of God'" (v.62; my emphasis vis a vis Elisha's activity at the time he was anointed by Elijah).
Inquiring minds might want to know if there's a similarity between Elijah's passing of his mantle to Elisha and Jesus' invitation to the two men, above, to follow him. If there is, the connection is likely coincidental, but the principle the three calls may have in common is this: When calling people, God does not impress them into service the way a draft board impresses men into the military.
Whether a person chooses to obey God's call or not is up to him or her. If, however, the person being called is willing to count the cost, so to speak, and is thereafter submissive to God's leading, she or he will likely heed the call willingly, gladly, and then be at peace with the decision, at least initially.
In a sense, then, Elijah's "what have I done for you?" put the ball into Elisha's court. Unlike the would-be followers of Christ who wanted to bury their father (i.e., wait until the father died!) or who wanted to say good-by to the people at home before becoming Jesus' disciples, Elisha, after kissing his parents goodbye, then followed Elijah and "ministered to him" (v.21b).
Moreover, realizing he no longer needed the oxen he was plowing with before his "call," he sacrificed them, parboiled their meat with a fire fueled by the plowing implements he no longer needed, and then threw a party for his household. The main course: ox meat (see v.21).
In conclusion, I think we are safe in concluding that Elisha, an obviously wealthy (and hardworking) land owner, did in fact count the cost of being Elijah's prophet in training. After doing so, he gladly submitted himself to the aging and discouraged Elijah (see 1 Kings 19:1-18, esp. v.4). Nor did Elisha start out his apprenticeship by performing spectacular miracles. Later in 2 Kings 3:11, a servant of King Jehoshaphat describes Elisha as a prophet who "used to pour water on the hands of Elijah"! In other words, Elisha went from being a wealthy farmer to a mere servant of the prophet Elijah, whose greatness was defined not by wealth (at one point an angel had to supply him with food and drink!--see 1 Kings 19:5 ff.), but by his obedient relationship with the one true God: Yahweh, the LORD.