Will God destroy the earth again? A friend of mine once said that, "God said he would never destroy the earth again." I assume she was referencing the time of Noah's Ark, where the Great Flood wiped out the entire world. 

I was simply wondering if this was true, and if there were any other cases in the Bible (or elsewhere) in which God said something like this?
Yes, God did say something like that after the flood. You can find this in Genesis, however the reference is specifically to destruction by flood (emphasis added):


  Genesis 9:8-17 (ESV)
  Then God said to Noah and to his sons with him, “Behold, I establish my covenant with you and your offspring after you, and with every living creature that is with you, the birds, the livestock, and every beast of the earth with you, as many as came out of the ark; it is for every beast of the earth. I establish my covenant with you, that never again shall all flesh be cut off by the waters of the flood, and never again shall there be a flood to destroy the earth.” And God said, “This is the sign of the covenant that I make between me and you and every living creature that is with you, for all future generations: I have set my bow in the cloud, and it shall be a sign of the covenant between me and the earth. When I bring clouds over the earth and the bow is seen in the clouds, I will remember my covenant that is between me and you and every living creature of all flesh. And the waters shall never again become a flood to destroy all flesh. When the bow is in the clouds, I will see it and remember the everlasting covenant between God and every living creature of all flesh that is on the earth.” God said to Noah, “This is the sign of the covenant that I have established between me and all flesh that is on the earth.”


There will be a time when the earth is destroyed again, this time by fire. You can find one (of many) references to this in the New Testament (emphasis added):


  2nd Peter 3:4-10 (ESV)
  They will say, “Where is the promise of his coming? For ever since the fathers fell asleep, all things are continuing as they were from the beginning of creation.” For they deliberately overlook this fact, that the heavens existed long ago, and the earth was formed out of water and through water by the word of God, and that by means of these the world that then existed was deluged with water and perished. But by the same word the heavens and earth that now exist are stored up for fire, being kept until the day of judgment and destruction of the ungodly. But do not overlook this one fact, beloved, that with the Lord one day is as a thousand years, and a thousand years as one day. The Lord is not slow to fulfill his promise as some count slowness, but is patient toward you, not wishing that any should perish, but that all should reach repentance. But the day of the Lord will come like a thief, and then the heavens will pass away with a roar, and the heavenly bodies will be burned up and dissolved, and the earth and the works that are done on it will be exposed.