The idea of persevering is not just to "grin and bear it" but to remain under trials in a such a way that we glorify God as we learn the lessons the trials are meant to teach us, instead of seeking ways to get out from under (cf the prefix preposition "hupo" = under) the trials and be relieved of the pressure.

Hupomeno has the following meanings depending on the context - (1) Stay behind, to tarry behind (beyond an expected time), to remain (as in Acts 17:14, Luke 2:43). (2) To stand fast, endure or remain in the sense of persevering so that under affliction, trouble, opposition or trial one holds fast to one's belief or faith (Mt 10:22, 24:13, Mark 13:13, James 5:11, et al). The idea is to be patient under, to persevere and to do so bear bravely and calmly (from Thayer).

Wayne Detzler recounts an amazing true life example of Christian perseverance writing that...

True Christian perseverance is not tied to tenacity. It is rather the work of God the Holy Spirit in a believer's life. The starch in a saint's spine is shown by Scripture to be nothing less than the sanctifying work of the Holy Spirit. Only in this way can one explain the work of Gladys Aylward, a London parlor maid. Societies scorned her missionary application. She seemed too dull to master Chinese and fulfill her vision of serving in China. Realizing this, she scoured up her own fare to China and sailed in 1930. After slogging her way across Siberia she reached her field in remote Yangcheng. When the Japanese invaded in 1940 she led 100 children on an epic journey that caught the imagination of Hollywood (Ed: Watch the movie about her life - The Inn of the Sixth Happiness or DVD). In 1947 failing health forced her back to England where she crusaded for missions until her death in 1970. That was tenacity, not just British grit. It is God's persevering grace.  (Detzler, Wayne E: New Testament Words in Today's Language. Victor. 1986)

Another great example of a Christian who endured (persevered) under trials in a God honoring way is William Wilberforce (biography in Wikipedia or short bio in Christian History) the 19th-century parliamentarian, who was moved by the Lord to oppose the lucrative but humanly degrading slave trade. In 1807 Wilberforce brought about the banning of the slave trade in England but it was not until 1833 was slavery as an institution abolished, this news reaching Wilberforce even as he lay on his deathbed. Talk about persevering!

Spurgeon said

By perseverance the snail reached the ark.

Perseverance is also illustrated in nature for...

Today's mighty oak is just yesterday's little nut that held its ground

Coleman Cox offered another example from nature noting that...

Even the woodpecker owes his success to the fact that he uses his head and keeps pecking away until he finishes the job he starts.

Samuel Johnson claimed

Great works are performed not by strength but by perseverance.

William Secker put it well when he said that...

Though Christians be not kept altogether from falling, yet they are kept from falling altogether.

Hupomeno was a military term used of an army’s holding a vital position at all costs. Every hardship and every suffering was to be endured in order to hold fast.

Endurance is a critical Christian virtue. Unless we have endurance , we can never learn many of the truths that God wants us to learn, truths that will lead us into a deeper life and a more fruitful ministry. Children are usually impatient; they cannot sit still long enough to get the things done that need to be done. “How long do we have to wait?” is the stock question of the child. Impatience is a mark of immaturity. Impatience is also a mark of unbelief.

Trench defined hupomeno (hupomone) as manifesting the

temper of spirit in which we accept God’s dealings with us as good, and therefore without disputing or resisting.

Vine writes that dokimos signifies...

that which is approved by being proved, that which stands the test  (Vine, W. Collected writings of W. E. Vine. Nashville: Thomas Nelson or Logos)

Wuest adds this description that dokimos means to...

put to the test for the purpose of being approved, and having met specifications, having the stamp of approval placed upon one. (Wuest, K. S. Wuest's Word Studies from the Greek New Testament: Eerdmans or Logos)

A dokimos man or dokimos character is like metal which has been cleansed of all alloy and impurity. In James 1:12 (see below) the weaknesses of such a one have been eradicated and he emerges strong and pure. That which is dokimos is shown to be trustworthy and genuine.

Approved describes anything tested and fit for service. As alluded to above, this term was used of gold and silver which has been purified by fire of all alloy.

Dokimos is the word describing money which is genuine or as we would say sterling (silver) [sterling = conforming to the highest standard]. In other words, a persons must first be "proved" before being "approved". One so approved is assayed by the One Who has eyes like flames of fire (see note Revelation 1:14) yet passes this scrutiny and is counted as worthy.

Dokimos is a word which motivates one to have a "God consciousness" (cp "Coram Deo" - before the face of God!), a consciousness of His presence and of living and acting in His sight, so as to please Him in all things. (e.g., see the use by James below)

Sometimes it is helpful to get a sense of the meaning of a word by observing uses of its antonym and here Isaiah 1:22 presents us with a clear picture, where God is speaking to faithless Israel declaring...

Your silver has become dross (Hebrew = siyg = literally  that which is turned away or skimmed off in the refining process, the waste or impurity, the refuse after smelting precious metal and figuratively that which is base or worthless), Your drink diluted with water. (Comment: The Septuagint -LXX translates siyg with the Greek word adokimos)

Richards writes that dokimos

is used in the NT in the sense of recognition, of being officially approved and accepted.

Barclay writes that...

The Greek for one who has stood the test is dokimos, which describes anything which has been tested and is fit for service. For instance, it describes gold or silver which has been purified of all alloy in the fire. It is therefore the word for money which is genuine, or, as we would say, sterling. It is the word used for a stone which is fit to be fitted into its place in a building. A stone with a flaw in it was marked with a capital A, standing for adokimastos, which means tested and found wanting. Timothy was to be tested that he might be a fit weapon for the work of Christ, and therefore a workman who had no need to be ashamed. (Barclay, W: The Daily Study Bible Series. The Westminster Press or Logos)

The root dek-, dechomai, accept, gives two verbal derivatives dokeo and dokao. The former means (intrans.) to appear, have the appearance, (trans.) to think, believe, consider right; the latter means expect. Derivatives of the former are: (a) dokimos, trustworthy, reliable, tested, recognized, used as a technical term for genuine, current coinage, but also applied to persons enjoying general esteem; (b) adokimos, untested, not respected; (c) indirectly also dokimion, test, probation; (d) from dokimos are also derived dokimazo, test, pronounce good, establish by trial, recognize, and apodokimazo, disapprove of, reject, blame; dokimasis and dokimasia, investigation, testing (preparatory to installing in an office); dokime, approved character, trial.  (Brown, Colin, Editor. New International Dictionary of NT Theology. 1986. Zondervan) 

Spurgeon in Faith's Checkbook - Mark of Divine Approval - YES, he is blessed while he is enduring the trial. No eye can see this till he has been anointed with heavenly eye salve. But he must endure it, and neither rebel against God, nor turn aside from his integrity. He is blessed who has gone through the fire and has not been consumed as a counterfeit. When the test is over, then comes the hallmark of divine approval, “the crown of life.” As if the Lord said, “Let him live; he has been weighed in the balances, and he is not found wanting.” Life is the reward: not mere being—but holy, happy, true existence—the realization of the divine purpose concerning us. Already a higher form of spiritual life and enjoyment crowns those who have safely passed through fiercest trials of faith and love. The Lord hath promised the crown of life to those who love Him. Only lovers of the Lord will hold out in the hour of trial; the rest will either sink or sulk, or slink back to the world. Come, my heart, dost thou love thy Lord? Truly? Deeply? Wholly? Then that love will be tried, but many waters will not quench it, neither will the floods drown it. Lord, let thy love nourish mine to the end. (Spurgeon, C. Faith's Checkbook)

The stephanos was literally an adornment worn around the head as a crown of victory in the Greek athletic games, this reward being given to the runner who crossed the goal first, to the disc thrower with the longest toss, etc. Apart from recognition of athletes and winners of various kinds of competitions, in the Greco-Roman world,  the awarding of a crown or wreath signified appreciation for exceptional contributions to the state or groups within it. The recipients were usually public officials or civic-minded persons serving at their own expense

Barclay adds that stephanos had the following associations in the ancient world...

(a) the victor's crown in the games. Smyrna had annual games which were famous all over Asia. As in the Olympic Games, the reward of the victorious athlete was the laurel crown. The Christian can win the crown of victory in the contest of life.

(b) When a man had faithfully performed the work of a magistrate, at the end of his term of office he was granted a crown. He who throughout life faithfully serves Christ and his fellow-men will receive his crown. 

(c) The heathen world was in the habit of wearing crowns, chaplets of flowers, at banquets. At the end of the day, if the Christian is loyal, he will have the joy of sitting as a guest at the banquet of God.  

(d) The heathen worshippers were in the habit of wearing crowns when they approached the temples of their gods. At the end of the day, if he has been faithful, the Christian will have the joy of entering into the nearer presence of God.

(e) Some scholars have seen in this crown a reference to the halo or the nimbus which is round the head of divine beings in pictures. If that is so, it means that the Christian, if he is faithful, will be crowned with the life which belongs to God himself. (Barclay, W: The Daily Study Bible Series. The Westminster Press or Logos)

The stephanos was the only prize ancient Olympic athletes received and thus it was cherished as a great treasure. How much more should we as believers "run with endurance the race that is set before" (see note Hebrews 12:1) us, knowing that the Olympic athletes

do it to receive a perishable wreath (stephanos) but we an imperishable." (See note on 1Cor 9:25)

TDNT has a lengthy note on stephanos...