Here we have the first of three references to the valley of Gilgal near the end of Ether 13. Technically, the name Gilgal is a biblical name (it occurs 41 times in the King James Bible), but it is quite possible that this biblical connection was unrecognized by Oliver Cowdery and Joseph Smith. Initially spelled as Gilgall here in 𝓞, the name was later corrected to Gilgal (Oliver crossed out the final l with a much heavier ink flow). For this first occurrence of “the valley of Gilgal”, it appears that Oliver and Joseph decided to check the spelling. We should note here that this example clearly shows that Joseph must have spelled out the name letter by letter since orally he could not have otherwise distinguished between Gilgal and Gilgall. (A similar argument is made for the spelling out of the name Coriantumr in Helaman 1:15, as discussed under that passage.)
It is worth noting that the name Gilgal appears twice earlier in the text, the first time as the name of a city (in 3 Nephi 9:6) and the second time as the name of a general (in Mormon 6:14). In those two cases, 𝓞 is not extant, nor does 𝓟 show any variation in the spelling for Gilgal. The two subsequent spellings of Gilgal in Ether 13, in verses 29 and 30, also show no variation in 𝓟 (neither is sufficiently extant in 𝓞 to note any variation there).
Summary: Maintain throughout the Book of Mormon text the spelling of the name Gilgal, which is identical in spelling to the biblical name.