Re-baptisms

 I know that in my traveling and preaching, many a time I have stopped by beautiful streams of clear, pure water, and have said to myself, "How delightful it would be to me to go into this, to be baptized for the remission of my sins." When I got home, Joseph told me it was my privilege. At this time, came a revelation, that the Saints could be baptized and re-baptized when they chose, and then that we could be baptized for our dear friends. (Brigham Young, J.D. 18:241)

I would be baptized seven times, were I in your place; I would not stop teasing some good Elder to baptize me again and again, until I could think my sins forgiven. I would not live over another night until I was baptized enough to satisfy me that my sins were forgiven. Then go and be confirmed, as you were when you first embraced the religion of Jesus. That is my counsel. (Brigham Young, J.D. 2:8, 9)

So there were many reasons (not including excommunication)  a member of the LDS church would get re-baptized in the early days of Joseph and Brigham.

• Rededication and renewal
• On this day the Twelve were re-baptized. Why? Because the Church, having broken old ties in the East was, in a way, experiencing a new birth. Because, owing to conditions of life on the plains, regular Church routine could not always be observed. For this reason for non-observance of certain regulations were made by the people and accepted by their leaders. But now those who stood at the head of the Church wanted a gesture of support to themselves and a sign that willing obedience would be given to the rules of the Church. This was affected by re-baptism. (Wilford Woodruff Journal, August 6, 1847)
• That seems to be a kind of standing ordinance for all Latter-day Saints who emigrate here, from the First Presidency down: all are re-baptized and set out anew by renewing their covenants. (Orson Pratt, J.D. 18:160)
• The Saints wore more lively there than I could have expected, they being visited very seldom by any of the Elders. They were ready, many, for re-baptism but I could not attend to it then, and thought it best to let them consider upon it and the more get ready to go at once. Now it was almost a general thing through England [29] that the Saints were being re-baptized, for they had many and mostly become old and cold and it required a renewal of covenants and fresh works together with mere faith and diligence, to give the work new impetus and revive the dropping spirits of the Saints and the work generally. (Huntington Diary, Feb. 7, 1847, p. 114)
• Wikipedia article on Jedediah Grant (Heber Grant's father): In 1856, Grant was called upon by Young to tour the northern sections of Utah, calling the Latter-day Saints to repentance. Grant inspired the Mormon Reformation of 1856 as he delivered fiery speeches on this tour. He issued a call for rebaptism of all the members of the area. Grant's speeches earned him the title, "Brigham's Sledgehammer." The effects of his speeches were felt almost immediately; members throughout the area, as well as in distant parts, were rebaptized to signify their commitment to renew their commitments to the LDS Church and the gospel. Several of these speeches are recorded in Journal of Discourses.Pasted from <http://en.wikipedia.org/wiki/Jedediah_M._Grant> 
• Be humble, and seek unto the Lord as children; put away your vain philosophies and strip yourself for the Kingdom. Preach Christ, and him crucified--preach life and salvation unto the Elders; the Saints [38] first, and then unto the people. Begin at home in the office--purify and regenerate them; get the fulness of the Spirit, and the power of the Highest to rest down upon you and then go forth to the people, and let a reformation be stirred up among the Saints; fire up each other, and then all the Saints; let all participate, and when they have sufficiently cleansed the inside of the platter, let them cleanse the outside, and renew their covenants in the waters of baptism, and abide in the truth, and be alive in the Church and Kingdom of God. (Letter of Brigham Young to Orson Pratt in England, Mill. Star, 19:97)
• Requirement to enter United Order
• December 30, 1875--This afternoon Mary, Fanney and myself were baptized for the remission of sin and renewal of our covenants and for the observance of the rules of the United Order, under the direction of L. John Nuttall, who administered the Ordinance of Baptism. (Diary of J. H. Standifird)
• Having been commissioned of Jesus Christ, I baptize thee for the remission of thy sins, for the renewal of the covenant with God and thy brethren and for the observance of the rules that have been said in thy hearing, in the name of the Father and the Son and of the Holy Ghost, Amen. (Ceremony of baptism into the United Order; obtained this copy from among the papers of Apostle C. C. Rich, Sept. 4, 1886. The copy from which the above was taken is in the handwriting of George Q. Cannon.) (Unpublished Diary of Milando Pratt)
• Prerequisite for Temple ordinances (marriage)
• September 17th 1886--As it was customary to get baptized to prepare for my temple [48] ordinances, before going to the temple so as to be free from all evil and wrong... I arranged with Brother Leatham, who has charge of baptisms on the Temple Block, to be baptized. . . I had already made arrangements to get Ida's recommend to be baptized and at 2 p.m. . . . . I took Ida to the Old Endowment House and after a word of prayer and a few remarks by Brother Leatham, he baptized us and confirmed us for the renewal of our covenants. (Diary of John M. Whitaker, Book 3, p. 16.)
• It was customary in those days to be re-baptized before being married. This young couple adhered to that practice, though one foot of ice in big creek had to be broken in order to do so. (Life of George F. Richards, p. 8)
• For Healing
• Tues. 4, 1842, Emma is very sick again. I attended with her all the day, being somewhat poorly myself. Wed. 5. My dear Emma was worse. Many fears were entertained that she would not recover. She was baptized twice in the river, which evidently did her much good. She grew worse again at night, and continued very sick indeed. I was unwell, and much troubled on account of Emma's sickness. (History of Church, J. Smith, Vol. 5; p. 167-168)
• Baptisms for the dead, and for the healing of the body must be in the font, and those coming into the Church and those re-baptized may be baptized in the river. (D.H.C. 4:586)
• Upon arriving at Pisgah, Lorenzo and members of his family remained for a season. "At this place," Lorenzo said, "I was taken seriously and dangerously ill with a burning fever, which so affected my brain that I was delirious many days, lying at the point of death. While in this condition, Elder Phineas Richards, the father of Apostle F. D. Richards, assisted by other kind brethren, took me from my bed, wrapped in a sheet placed me in a carriage, drove to a stream of water, and baptized me in the name of the Lord, for my recovery. The fever immediately abated, and through the kind unwearied nursing and attention by my faithful, loving wives, and my dear sister, E.R.S. Smith, aided and sanctified through the power and blessing of God, I was delivered from suffering and restored to health. The sickness was the result of extreme hardships and exposures consequent on the journey. (Life of L. Snow, 1846, Romney, p. 80-81)
• Mrs. Richards was taken by her brother from a sick bed to a lake from the surface of which ice more than a foot thick [53] had been removed, and there baptized, whereupon she immediately recovered. Similar cases might be given by the score. (Bancroft's History of Utah, p. 337)
• Remission of Sins
• I was baptized first in 1832, and I was baptized the next time when I came out here to Utah. I was baptized first for the remission of sins. Was baptized the second time for the same thing. I came out here in 1847 and was re-baptized then. When I came out here I was baptized again renewing my covenants and also for the remission of sins that I might have committed in taking this long and tedious journey through all these mountains and canyons. (Joseph C. Kingsbury, Temple Lot Case, p. 340)
• My counsel to them today is, as it has been on former occasions to all who have [58] come into these valleys, Go and be baptized for the remission of sins, repenting of all your wanderings from the path of righteousness, believing firmly in the name of Jesus Christ, that all your sins will be washed away. If any of you inquire what is the necessity of your being baptized, as you have not committed any sins, I answer, it is necessary to fulfill all righteousness.  I have heard some of you cursing and swearing, even some of the Elders of Israel. I would be baptized seven times, were I in your place; I would not stop teasing some good Elder to baptize me again and again, until I could think my sins forgiven. I would not live over another night until I was baptized enough to satisfy me that my sins were forgiven. Then go and be confirmed, as you were when you first embraced the religion of Jesus. That is my counsel. (Brigham Young, J.D. 2:8, 9)
• In the afternoon, I attended a council of the Twelve at President Young's. There were present: Brigham Young, Heber C. Kimball, Orson Hyde, Orson Pratt, John Taylor, Wilford Woodruff, George A. Smith, Willard Richards and Brother Hyrum Smith. . . . This Council was called to re-consider the case of Orson Pratt, who had previously been cut off from the quorum of the Twelve for the neglect of duty; and Amasa Lyman had been ordained an Apostle in his place. I told the Council that as there was not a quorum present when Orson Pratt's case came up before them, that he was still a member, that he had not been cut off legally, and I would find some other place for Amasa Lyman to which the Council agreed. President Young said there were but three present when Amasa was ordained. I told them that was legal when no more could be had. * * *  At three o'clock, Council adjourned to my house, and at four I baptized Orson Pratt and his wife, Sarah Marinda, and Lidia Granger in the Mississippi River, and [62] confirmed them in the Church, ordaining Orson Pratt to his former office and standing in the Quorum of the Twelve. (History of Joseph Smith, Mill. Star, Vol. 20:423)
• In the first place, if you were re-baptized for the remission of sins, peradventure you may receive again the Spirit of the Gospel in its glory, light and beauty; but if your hearts are so engrossed in the things of this world, that you do not know whether you want to be re-baptized or not, you had better shut yourselves up in some canyon or closet, to repent of your sins, and call upon the name of the Lord, until you get His spirit. (Brigham Young, J.D. 1:324)

Further reading:

D Michael Quinn, "The Practice of Rebaptism at Nauvoo"

https://ojs.lib.byu.edu/spc/index.php/BYUStudies/article/viewFile/5015/4665