I frequently visit Fulfilled Prophecy. I agree with this person's thoughts from the discussion room on what it means to overcome.
"Saving faith is enduring faith. Over and over again in Jesus' letters to the churches in Revelation He tells them to overcome and gives specific promises of blessings to the ones who actually do so. Jesus also warned of the great apostasy which would come upon the church in the last days - betrayals, misleading, hating, love growing cold - but to the "one who endures to the end, he will be saved" (Matt. 24:10-13). To overcome is to have enduring faith - faith that goes beyond belief into action (James - faith without works is dead). From Jesus' teaching we know what is in store - there are no excuses for being surprised or unprepared. The one who overcomes heeds the warnings of Jesus and by faith is willing to endure and not cave to the phenomenal persecution which lies ahead.
I don't think Jesus is hoping that some of us can step up and display this kind of faith. He knows to whom He has given this gift of faith and He knows that in the last days, when so-called believers are falling like flies, that true believers will readily display enduring faith because He is the One who has enabled us to do so.
I also don't think that "overcome" means "survive"- an overcomer's measure of success is not in physical survival but in spiritual steadfastness..."
John spoke of overcoming...
"You are from God, little children, and have overcome them; because greater is He who is in you than he who is in the world." 1 John 4:4
"For whatever is born of God overcomes the world; and this is the victory that has overcome the world--our faith." 1 John 5:4
"And who is the one who overcomes the world, but he who believes that Jesus is the Son of God?" (1 John 5:5)
"And they overcame him because of the blood of the Lamb and because of the word of their testimony, and they did not love their life even to death." (Revelation 12:11)
Those who are born of God have been gifted with enduring faith. All true believers will be overcomers. It's just as the commenter said, overcoming does not mean surviving physically, but overcoming is related to spiritual steadfastness, even when facing death.
Romans 8:29-39 tells us that God's saving power is greater than the coming phenomenol persecution:
"For whom He foreknew, He also predestined to become conformed to the image of His Son, that He might be the first-born among many brethren; and whom He predestined, these He also called; and whom He called, these He also justified; and whom He justified, these He also glorified. What then shall we say to these things? If God is for us, who is against us? He who did not spare His own Son, but delivered Him up for us all, how will He not also with Him freely give us all things? Who will bring a charge against God's elect? God is the one who justifies; who is the one who condemns? Christ Jesus is He who died, yes, rather who was raised, who is at the right hand of God, who also intercedes for us. Who shall separate us from the love of Christ? Shall tribulation, or distress, or persecution, or famine, or nakedness, or peril, or sword? Just as it is written, FOR THY SAKE WE ARE BEING PUT TO DEATH ALL DAY LONG; WE ARE CONSIDERED AS SHEEP TO BE SLAUGHTERED. But in all these things we overwhelmingly conquer through Him who loved us. For I am convinced that neither death, nor life, nor angels, nor principalities, nor things present, nor things to come, nor powers, nor height, nor depth, nor any other created thing, shall be able to separate us from the love of God, which is in Christ Jesus our Lord."
If you are one of God's elect, you will overwhelmingly conquer, you will be an overcomer, through Him!