The original text of the Book of Mormon sometimes has the archaic use of mine (and thine) rather than my (and thy) before vowel-initial words. In this particular example of mine arm, the 1841 typesetter accidentally replaced the archaic use of mine with my, yet earlier in this verse he correctly set mine arm (“notwithstanding I shall lengthen out mine arm unto them from day to day”). The 1849 LDS edition replaced the 1841 my arm with the original mine arm. For further discussion of this characteristic of the biblical style in the Book of Mormon text, see possessive pronouns in volume 3.
Summary: Maintain each archaic use in the Book of Mormon of mine (and thine) before vowel-initial words, providing it is supported by the earliest textual sources.