Initially, when copying from 𝓞 into 𝓟, Oliver Cowdery thought “this great sin” was conjoined with the preceding “the cause of his iniquity”; thus he accidentally repeated the preposition by (“by the cause of his iniquity and by this great sin”). But “this great sin” actually conjoins with the following “his many other sins” to form the subject for a separate clause (“and this great sin and his many other sins did harrow up his mind”). Almost immediately Oliver caught his error here in 𝓟 and crossed out the intrusive by (there is no change in the level of ink flow for the crossout). The original manuscript undoubtedly read as corrected in 𝓟.
In this verse, the editors for the 1920 LDS edition replaced the analytic “by the cause of ” with the expected “because of”. In contrast to 579 examples of “because of ” in the original text (with possible variation in the count depending on whether the original text had the of or not), there is one other case of “by the cause of ” in the original text of the Book of Mormon:
In addition, there is a passage with the phrase “by this cause” (instead of “because of this”) in Ether 7:25: “and by this cause the people were brought unto repentance”. There are also examples where the preposition is for rather than by, as in Alma 60:32: “while your iniquity is for the cause of your love of glory and the vain things of the world”. Thus the relatively rare use of “by the cause of ” (in Alma 7:5 and Alma 15:3) appears to be fully intended.
Historically, because derives from by cause (see under because in the Oxford English Dictionary). The more analytical but rarer expression “by the cause of” is found from Early Modern English up to the present, as in these examples (with accidentals regularized) from Literature Online
:
The critical text will therefore follow the earliest reading for the two original instances of “by the cause of ”, here in Alma 15:3 and earlier in Alma 7:5.
Summary: Maintain the original reading “by the cause of ” in Alma 7:5 and Alma 15:3.