In my mind as a child, I had not thought of what that power looked like. How is such power exercised? D&C 121:36 says, "That the rights of the priesthood are inseparably connected with the powers of heaven, and that the powers of heaven cannot be controlled nor handled only upon the principles of righteousness."
It goes on to list the attributes of those with power from God and the qualities of those that do not have His power.
This gives me a possible picture of what kind of people Nephi and Lehi were. They were long-suffering and gentle and meek. They loved authentically and were kind and had a pure understanding of the gospel. Their interactions with others enlarged each other's souls.
This reminds me of the quote from Joseph.
Nothing is so much calculated to lead people to forsake sin as to take them by the hand, and watch over them with tenderness. When persons manifest the least kindness and love to me, O what power it has over my mind, while the opposite course has a tendency to harrow up all the harsh feelings and depress the human mind. (Joseph Smith, Nauvoo Relief Society Minute Book, p. 59, History of the Church, 5:23–24.)It was the love of God and their fellow man that brought them so much power. Power to be able to influence the hearts of their friends and family to turn toward God.