For the 1837 edition, Joseph Smith replaced the preposition beside with by the side of. Historically, the word beside derives from “by side” (Oxford English Dictionary), so the substitution of by the side of for beside in 1 Nephi 2:6 can be viewed more as a type of stylistic editing than as a change in meaning. There appears to be no grammatical and semantic reason for the change. Such stylistic changes for the 1837 edition appear chiefly in the first book (1 Nephi). It should also be noted that it is the tent that is beside the river of water, not the valley. The substitution of by the side of does not clear up this potential ambiguity.
There are no other occurrences in the Book of Mormon text of either beside or by the side of. In the King James Bible, we have examples of both in describing location by water, although most cases have beside:
Note that the one example involving pitching of tents (Judges 7:1) uses beside. Other examples involving by the side have genitive modifiers (such as “by the river’s side” in Exodus 2:5 and Numbers 24:6).
Summary: The use of the original beside in 1 Nephi 2:6 is perfectly acceptable; the use of beside is supported by examples from the King James Bible and is not improved by replacing beside with the expansive, paraphrastic by the side of.