The Book of Mormon text reads Gazelem in all the textual sources, including the original manuscript, although the last vowel of the name has sometimes been spelled differently in other texts (described below).
It is difficult to tell here whether Gazelem is the name of the servant or the stone. The 1830 typesetter placed commas around the phrase a stone. One could interpret this punctuation as forcing Gazelem to be associated with the preceding servant (thus Gazelem is the name of the servant). Or one could interpret a stone as a nonrestrictive appositive, thus describing Gazelem as a stone.
Joseph Smith adopted Gazelem as one of his code names in a number of sections in the first published edition of the Doctrine and Covenants (in 1835). Orson Pratt’s explanation of why this name was chosen for Joseph suggests the interpretation that Gazelem was the name of a person (original accidentals here retained):
The name here was spelled as Gazelum. And the code name in the 1835 Doctrine and Covenants was spelled yet another way, as Gazelam (six times). For additional discussion on the use of code names in the original Doctrine and Covenants, see David J. Whittaker, “Substituted Names in the Published Revelations of Joseph Smith”, Brigham Young University Studies 23/1 (1983): 103–112.
There is also evidence that the name Gazelem could be the name of the stone. Assigning a name to an instrument of revelation is supported in verse 38 where the ball or compass that Lehi found is referred to by its name Liahona. One example of interpreting Gazelem as the name of the stone is found in the index for the 1841 British edition (which was printed under the direction of Brigham Young, Heber C. Kimball, and Parley P. Pratt); here we have the following list of objects (plates and instruments of translation) that Alma handed over to Helaman:
The index for the 1841 edition, produced in Britain at the time of the publication of that edition, was created independently and without any input from Joseph Smith. Subsequent derivative indexes made for the earlier printed 1837 and 1840 editions repeat the same identification of Gazelem as a stone. On the other hand, the index for the 1830 edition (which was not produced at the time of the 1830 publication) ignores the stone and refers only to the directors (“The directors spoken of ”). Most likely, the phrase here in Alma 37:23, “my servant Gazelem”, intends to say that the name of the servant was Gazelem. One reason is that there are numerous instances in the text of the expression “my servant ”:
Another reason is that if the name of the stone were Gazelem, we would expect the word order to be different, with the indefinite a stone preceding its name (something like “I will prepare unto my servant a stone, Gazelem”). For example, names like Rameumptom and Liahona are given only after the object itself has been described or identified:
The critical text will therefore interpret the name Gazelem as the name of the servant, not the name of the stone.
Summary: In Alma 37:23, the name of the servant is spelled Gazelem; based on usage elsewhere in the text, there is less chance that Gazelem is the name of the stone.