The usage their names is rather strange here since this noun phrase is postmodified by a relative clause, “which did contend for the judgment seat”. The 1840 edition replaced the their with a the, which is what we expect in modern English. The 1908 RLDS edition restored the original their to the RLDS text.
Usage elsewhere in the text supports the the when the plural noun names is postmodified; there are 15 cases, of which six are of the form “the names <<em>of-phrase> ”:
On the other hand, the earliest text has 11 instances of their names, of which only this one here in Helaman 1:3 has any postmodification, namely, a relative clause without an of-phrase: “their names which did contend for the judgment seat”. One could argue that an original the names was here changed to their names in the early transmission of the text because one expects something like “their names were X, Y,.. .”, as in these examples:
On the other hand, one could argue that if the names was correct, then we should expect “the names of those which did contend for the judgment seat”. These opposing expressions suggest that what we have here in Helaman 1:3 is a unique but intended reading in the text. The critical text will therefore maintain the reading of the earliest text, “their names which did contend for the judgment seat” (the reading in 𝓞).
Summary: Maintain the earliest reading in Helaman 1:3, “these are their names which did contend for the judgment seat”, the reading of the original manuscript.