As noted by Breitbart, her actions have not affected only her, but the staff under her responsibility as well, which is the true reason why this case has had any attention at all.
Breitbart: Supreme Court Denies Relief to Christian County Clerk—For Now
A "money quote" from the article reads (emphasis added):
"...she is both claiming a personal right based on her religious belief, but then also claims the right to use her governmental power to order other civil service to act only as she acts, even if they do not share her religious objection."
This is wrong, and not supported by biblical narrative as to how one who follows God is supposed to face persecution, let alone conduct themselves in ordinary manners.
Many, including a politician I have liked, have now been quick to claim religious persecution based on her being sent to jail for contempt of court. This action is rash and does not reflect biblical Christianity, and only serves to remove the salt from our witness.
I'll cite first off a sermon of Martin Luther King Jr. on "But if Not", the story from Daniel 3 about Shadrach, Meshach, and Abed-Nego. If you've not heard this sermon, you need to hear it, especially given the social context the sermon was given then in 1967, and the circumstances we find ourselves in today.
The contrast between this sermon and the calls today is that the motivation is a personal conviction. A deep conviction that moves you to stand up, even to the point of death. Civil disobedience then is something that comes from a deep personal conviction, something that you yourself are completely dedicated to. It does not come from someone else.
Note that when King references those who are joining him in his civil disobedience, he does not reference any of his contemporaries. He does not reference those whom he has forced to join in his civil disobedience. His call in that sermon is to his contemporaries to join him based on a shared personal conviction. His call is to have a deep personal conviction.
In the story of Shadrach, Meshach, and Abed-Nego, where do they prevent others from worshiping the idol? Who does God say is joining them in the fire? It's God. Not contemporaries, or those under their responsibility. This is seen in the chapter just prior, in that Daniel and his friends were placed in positions of great responsibility in Babylon.
Yet the complaint to King Nebuchadnezzar is only about these three, and not their subordinates. The Babylonians were all obedient to the law of the land, and these three stood alone in their disobedience.
Though as Martin Luther King Jr. points out, they were not alone, as they had the companionship of God. It was by that companionship that they were saved, though they were still thrown in the fire. The material consequences of breaking man's law were still enacted, but by the grace of God, they were able to endure it and continue on.
This is a theme that is littered throughout the Bible. Suffering is an isolating experience to those who do not have God, but for those who have God, suffering has a rather different connotation.
"And not only that, but we also glory in tribulations, knowing that tribulation produces perseverance; and perseverance, character; and character, hope. Now hope does not disappoint, because the love of God has been poured out in our hearts by the Holy Spirit who was given to us." - Romans 5:3-5 (NKJV)
"My brethren, count it all joy when you fall into various trials, knowing that the testing of your faith produces patience. But let patience have its perfect work, that you may be perfect and complete, lacking nothing. " - James 1:2-4 (NKJV)
"We are hard-pressed on every side, yet not crushed; we are perplexed, but not in despair; persecuted, but not forsaken; struck down, but not destroyed - always carrying about in the body the dying of the Lord Jesus, that the life of Jesus also may be manifested in our body." - 2 Corinthians 4:8-10 (NKJV)
Our sufferings will come, but they do not do so in vain.
"But may the God of all grace, who called us to His eternal glory by Christ Jesus, after you have suffered a while, perfect, establish, strengthen, and settle you." - 1 Peter 5:10 (NKJV)
"For I consider that the sufferings of this present time are not worthy to be compared with the glory which shall be revealed in us." - Romans 8:18 (NKJV)
But when suffering to the glory of God, there are limits and guidelines.
"If you are reproached for the name of Christ, blessed are you, for the Spirit of glory and of God rests upon you. On their part He is blasphemed, but on your part He is glorified. But let none of you suffer as a murderer, a thief, an evildoer, or as a busybody in other people’s matters. Yet if anyone suffers as a Christian, let him not be ashamed, but let him glorify God in this matter." - 1 Peter 4:14-16 (NKJV)
Even intentional activities we would desire to partake in, such as fasting or prayer, are private and personal matters.
"And when you pray, you shall not be like the hypocrites. For they love to pray standing in the synagogues and on the corners of the streets, that they may be seen by men. Assuredly, I say to you, they have their reward. But you, when you pray, go into your room, and when you have shut your door, pray to your Father who is in the secret place; and your Father who sees in secret will reward you openly." - Matthew 6:5-6 (NKJV)
"Moreover, when you fast, do not be like the hypocrites, with a sad countenance. For they disfigure their faces that they may appear to men to be fasting. Assuredly, I say to you, they have their reward. But you, when you fast, anoint your head and wash your face, so that you do not appear to men to be fasting, but to your Father who is in the secret place; and your Father who sees in secret will reward you openly." - Matthew 6:16-18 (NKJV)
The point being that the foundation of your faith, your personal convictions, are just that, personal. They are not suitable to be enforced on others, nor done to achieve personal gain or recognition, as those are immediately rewarded.
With Kim Davis, she is not just acting on her own personal convictions, and the lawsuit is not that she is allowing her subordinates to issue licenses but she refuses, but that she is forcing others into a fire which they cannot withstand because they may not have the companionship of God.
How cruel would it have been for Shadrach, Meshach, and Abed-Nego to have forced all those who were subordinate to them also suffering in the fires, albeit they without the grace of God to help them endure? While it is certainly not their fault that others were indeed consumed in the process of carrying out the orders of the King Nebuchadnezzar, the three men in their civil disobedience took responsibility only for themselves and their own personal convictions.
It is for this reason that Kim Davis is wrong, and those who support categorizing her behavior as religious persecution only do damage to efforts of evangelism. Their actions are not unlike that of Muslims in recent history, refusing to do their jobs on claims of religious persecution, but then instead of their faith carrying them through the trial, they use the legal system to get recourse and relief. They are not willing to give up their life or job for their faith, so why aren't Christians behaving differently?
"And whoever does not bear his cross and come after Me cannot be My disciple." - Luke 14:27 (NKJV)
"He who finds his life will lose it, and he who loses his life for My sake will find it." - Matthew 10:39 (NKJV)
No believer has the right or permission to force someone else lose their life for a relationship they don't already have. We are certainly called to witness to these people, and that we live out the effects of the foundation of our personal convictions, but at no time or place are we to enforce that on others.
"Therefore whatever they tell you to observe, that observe and do, but do not do according to their works; for they say, and do not do. For they bind heavy burdens, hard to bear, and lay them on men’s shoulders; but they themselves will not move them with one of their fingers. But all their works they do to be seen by men. They make their phylacteries broad and enlarge the borders of their garments...And whoever exalts himself will be humbled, and he who humbles himself will be exalted." - Matthew 23:3-5, 12 (NKJV)
Know that trying to replace the grace of God with laws of man to endure suffering will only delay the inevitable and invite the unintended. It is your faith, your dependence on God's grace and mercy, that should sustain you, be what gets you up in the morning, and has you sharing your convictions.
Pursue civil disobedience, but do it out of those personal convictions. You may stand alone so far as man is concerned, but you truly stand with God when you are persecuted for obedience to His commands instead of man's. You will still experience suffering, but in your suffering God is glorified, and you will be sustained in the midst of the fire.