The phrase to go the extra mile, and its variants, mean: to try especially hard to achieve something or do it well.

This phrase originally occurred as to go the second mile, in allusion to the account of Christ’s Sermon on the Mount in the gospel of Matthew, 5:41.—The gospel of Matthew, 5:38-41, is  as follows in the King James Bible (1611):

38 Yee haue heard that it hath beene said, An eie for an eie, and a tooth for a tooth.
39 But I say vnto you, that yee resist not euill: but whosoeuer shall smite thee on thy right cheeke, turne to him the other also.
40 And if any man will sue thee at the law, and take away thy coate, let him haue thy cloake also.
41 And whosoeuer shall compell thee to goe a mile, goe with him twaine.

The allusion to the gospel of Matthew, 5:41, is explicit in the two texts containing the earliest occurrences of to go the second mile that I have found. These two texts are as follows, in chronological order:

1-: Letter IV. The ancient Law of Retaliation abolished by the Gospel, in Letters addressed to Caleb Strong, Esq. late Governor of Massachusetts, showing, that retaliation, capital punishments, and war, are prohibited by the Gospel; justified by no good principle; not necessary to the safety of individuals or nations: but, incompatible with their welfare; inconsistent with the Christian character; and contrary to the laws of Christ (New York: Published by Dodge and Sayre, 1816), by the American Presbyterian preacher and religious writer Samuel Whelpley (1766-1817):

“And whosoever shall compel thee to go a mile, go with him twain.” You will be aware, Sir, that the bold, liberal, and impressive oriental style is here used, as in the case of the coat and cloak. It will certainly not be understood, that the Christian, when some of his property is taken away, is of his own accord to double she sum; or that when his liberty is restrained awhile, he must voluntarily protract or double the term of his duresse: far from it. If that would be overstraining the generous and ardent style of the Evangelist, which as much disdains the pedantic monotony of cold criticism, as it tramples in the dust the pride and selfishness of human ambition; how does that Christian obey Christ’s law, who is quick to resent injury and repel force by force; who, when his property is invaded, only abandons the pursuit of redress, far within the invader’s territory, crowning his full recovery with damages and cost, and sweetening his triumph with revenge; and who, when his liberty is assailed, instead of going the second mile, would pour out the last drop of his own blood, and that of his assailant, rather than go one rod? *

[* Here, the noun rod denotes a unit of length equal to 5½ yards, 16½ feet (approximately 5.03 metres).]

2-: To the Saints, an unsigned text published in the Deseret News (Salt Lake City, Utah, USA) of Saturday 6^th March 1852:

And Jesus taught on this wise: If any man shall compel you to go a mile, go with him twain; and if any man shall sue you at the law and take away your coat, let him have your cloak also.
[…]
Jesus did not say, if a saint, a brother, a friend compel you to go a mile, go with him two; no! but if any man, saint or sinner; for saints are subject to vanity and folly, and do wrong sometimes; and when exercised by a foolish spirit, are as likely as other men to compel you to travel with them, if they have the power to do it; or, take away your coat at the end of a law-suit; or do anything else, which might gratify the foolish spirit which has dominion over them, by vexing you, and causing anger to spring up in your hearts, because of which the Spirit of light, love, and wisdom, which you delight in, would take its departure from your souls.
Thus far, then, the spirits of evil have accomplished their designs with you; you have given way to temptation, and the Holy Ghost has taken its departure; and why? Because you could not take the spoiling or robbing of your coat joyfully, and send your cloak along with it as a witness of your love of the truth, over and above your love for worldly goods; you could not go the second mile, when you had been forced to go, because you [several illegible words] your fence making, or mending; your flocks or your mill tending; but if you had gone the second mile voluntarily, and preached righteousness to the man who compelled you to go the first mile, how do you know but that he would have received the truth, been converted to the faith of Jesus, and returned with you, not the two miles only, but traveled with you all your days, helped you to repair all the losses you had suffered on his account, and become a co-worker with you forever, in building up God’s Kingdom?

Likewise, the allusion to the gospel of Matthew, 5:41, is explicit in the two texts containing the earliest occurrences of to go the extra mile that I have found. These two texts are as follows, in chronological order:

1-: From the Cambridge Chronicle (Cambridge, Massachusetts, USA) of Saturday 4^th May 1901:

REV. DANIEL EVANS’S DISCOURSE.

Rev. Daniel Evans’s discourse, Sunday evening, was from Matt., 5-41: “Whosoever shall compel thee to go a mile, go with him twain!” In this verse we discover the compulsory and the voluntary rule of life. We find it illustrated in the family. The husband must provide for his household; the wife must do all things for his comfort, and both are governed by the outward pressure of public opinion which is compulsory and rigid in its demands. But beyond all that, love steps in and leads to all those little amenities which are just as sweet and precious after marriage as before, and we “go the extra mile” in fulfilling love’s demands.
Children, too, may be compelled to obey and conform to the rules of the house, but there should also be the glad service when they seek new ways of showing devotion.
In society, too, we must be civil to others; that is compulsory, a duty required of all in intercourse with others. But with this we may also be courteous, showing the spirit of Christ to others, winning by our own desires of helping them, and so we “go the extra mile,” by stepping beyond the formal requirements of society, and meeting others more than half way.
In business, too, tomorrow you must be in your place in office, store or workshop, when the bell rings or the whistle blows. At noon the signal comes for rest. He who works under the compulsory rules will drop his hammer even when uplifted for the final blow, his apron is off and hat on before the echoes have died away, and with a sigh of relief he is off. There is a way, a way of helpfullness [sic], a seeking to do soinething [sic] to lift the burden from other shoulders.
Mr. Evans told the story of a young man during the Spanish war. He noticed that his employer was overburdened with correspondence which was in Spanish. He knew that he could help him only by knowledge of the language, and so spent his evenings for four months in its study. Then he was ready for service, and offered his help, which was gladly accepted, of course. The difference is in being compelled to go a mile or doing so voluntarily.
Furthermore, we see the same law working in the religious life. We see some who seem to be doing just as little as possible if only they may secure their final acceptance and just enter the pearly gates. But there are those who join in the service, whose whole hearts are consecrated to Christ and his cause, who love much, and to them any extra service is welcome. They “go the extra mile” willingly, gladly.

2-: From The Standard Union (Brooklyn, New York, USA) of Monday 27^th July 1903:

SHOULD NOT BE CONTENT WITH AVERAGE IN LIFE
Dr. Ross, at Central Presbyterian, Says He Excels Who Does More Than Duty Requires.

The Rev. Dr. William Ross, of England, preached yesterday at the Central Presbyterian Church, Jefferson and Marcy avenues. Dr. Ross is in this country in connection with the Northfield Conference, where he is one of the speakers. He is the pastor of a large and influential Baptist church in Great Britain. Dr. Ross took his text from St. Matthew v., 41—“If they compel me to go a mile, I will go with them twain.”
“In connection with this text,” he said, “there are three thoughts I want to touch upon. First, average is not excellence; then, excellence is sacrifice; and, finally, sacrifice is victory. In the first thought we find the real essence of the Christian life. We must not be satisfied with the average in this world. It is easy enough to join the church, have faith in its doctrines and drift along with the crowd. But there is nothing in that to be looked upon as a great virtue. You can do that without effort or care. What is wanted in the Christian is that which is stated in the text—the going of the extra mile. Christ said to those around Him that unless their righteousness exceeded the righteousness of the Scribes and Pharisees, they could not enter the kingdom of heaven. The same is true to-day. Discipleship in Christ’s Church is not a passive or dormant state, but rather one in which we are called upon to push forward in the doing of good works.
“Some are good because they cannot help it. Over in my English church I have members who have been blessed exceedingly with all the riches of this life; they have never known trouble of any kind; yet they have never made a positive effort to further Christ’s kingdom. They are satisfied to be like the Scribes and Pharisees. There is no gradual but continual reaching upward toward the higher spiritual life; they never grow in closeness of communion to God. The man who approaches the Christian likeness of Christ is he who towers above his fellow men; it is he who is not satisfied with the average.
“Now, as to the second thought—that excellence means sacrifice. There is something more to do than simply to believe. We see the truth of this statement exemplified in the lives of the missionaries who go out to the barbarous, uncivilized countries to carry God’s words to the heathen. These noble men and women are not satisfied with inactive, passive subscriptions to certain articles of faith; they go the extra mile and tower above their fellow men. They are filled with the spiritual life that comes from living close to Christ and His teachings. They attain excellence, but they find that it brings its cares, its troubles and possible death. They have before them the great sacrifice made 1,900 years ago when Christ made the greatest sacrifice on the cross at Calvary.
“And now for the last thought bound up in our text—that sacrifice is victory. If you could only understand the great change that has come over the great dark continent of Africa since the American and English missionaries first pushed up the Congo and marked their progress by hundreds of graves from the coast to the interior you would then clearly see the truth of this last division of our text. But to rise above your fellow men, to be a force working for the regeneration of the world, to be able to make a sacrifice that counts, and, finally, to attain to a victory that is everlasting, it is necessary that you should follow the command and go the extra mile. Not faith alone, but practice, also, is needed. Continual practice finally becomes habitual, and when we have reached the stage where we work habitually for the advancement of Christ’s kingdom we have realized the meaning of our text.”