Quite simply, Abraham Lincoln got the nickname "Honest Abe" because ever since he was a young man in New Salem, Illinois, he was always incredibly honest. If he realized he had shorted someone a few cents, he would close the shop he worked at and deliver it.
Because of his honesty as a young man, he was often asked to judge contests and mediate arguments. He was clearly someone people were willing and able to trust, due to his unflinching honesty and notable trustworthiness.
Abraham Lincoln's honestly became rather infamous when he ran against Stephan Douglas for President. Douglas, after losing the race, noted, "You have nominated a very able and very honest man." Douglas, who had previously beaten Lincoln in a race for the Senate bid, clearly admired him for his candor.
Abraham Lincoln himself was vocal about his own honesty. He consistently made mention of the fact that he had never been deceitful in his politics. He is quoted as having said, "I am glad of all the support I can get anywhere, if I can get it without practicing any deception to obtain it." As this quotation seems to depict, he was very proud of his honesty and integrity.