The 1841 British edition replaced the nonassertive determiner any with the indefinite article a. The 1920 LDS edition restored the original any to the LDS text, by reference to one of the early editions. Either reading is, of course, possible. We have examples of both types elsewhere in the text, such as “neither is there any end of their treasures ... neither is there any end of their chariots” (2 Nephi 12:7, citing Isaiah 2:7) versus “neither hath he a shadow of turning from the right to the left” (Alma 7:20). In each case we follow the earliest textual sources.
Summary: Maintain the use of any in Mosiah 29:8: “neither should we have any right to destroy another” (the reading of the earliest textual sources).