I want to make it clear that I'm not really suggesting a "why" or "how" for these connections. Some people see nothing but plagiarism when they notice similarities between one book of scripture and another. I see it quite differently -- all of these words belong to God and the Book of Mormon claims to be a gathering of the word of God in one (see 2 Nephi 29:14). However you choose to view these connections, don't let them get in the way of appreciating the beauty of the language.
Tender Mercies
9 The Lord is good to all: and his tender mercies are over all his works. (Psalm 145)
20 ...But behold, I, Nephi, will show unto you that the tender mercies of the Lord are over all those whom he hath chosen, because of their faith, to make them mighty even unto the power of deliverance. (1 Nephi 1)
16 Hear me, O Lord; for thy lovingkindness is good: turn unto me according to the multitude of thy tender mercies. (Psalm 69)
8 And after I had traveled for the space of many hours in darkness, I began to pray unto the Lord that he would have mercy on me, according to the multitude of his tender mercies. (1 Nephi 8)
See also Ether 6:12. There are about a dozen additional references to "tender mercies" in the Psalms.
My Rock and My Salvation
2 He only is my rock and my salvation; he is my defence; I shall not be greatly moved. (Psalm 62)
36 And in them shall be written my gospel, saith the Lamb, and my rock and my salvation. (1 Nephi 13)
See also 1 Nephi 15:15 and 2 Nephi 4:30. In addition, see D&C 18:17 and Abraham 2:16.
In the Paths of Righteousness
3 He restoreth my soul: he leadeth me in the paths of righteousness for his name’s sake. (Psalm 23)
5 And it came to pass that they did humble themselves before the Lord; insomuch that I had joy and great hopes of them, that they would walk in the paths of righteousness. (1 Nephi 16)
19 For I perceive that ye are in the paths of righteousness; I perceive that ye are in the path which leads to the kingdom of God; yea, I perceive that ye are making his paths straight. (Alma 7)