This construction is somewhat unusual. We expect something more like “thy will be done”, as can be found in the King James version of the New Testament and elsewhere in the Book of Mormon:
In none of these is the noun will modified by an adjective such as righteous. Moreover, the adjective righteous here in Ether 12:29 is nonrestrictive in meaning. Moroni is saying that “thy will— which is righteous—be done”. He is not implying that there is such a thing as the unrighteous will of the Lord and that only the righteous will of the Lord should be done. In 2 Nephi 4:5, there is a similar nonrestrictive use of an adjective (namely, right) in the phrase “the right way that ye should go” (for discussion, see under that passage). Don Brugger also points out (personal communication) that there is the phrase “his holy will” in Moroni 7:2, where holy is used nonrestrictively.
Another possibility worth noting here in Ether 12:29 is that this expression, “thy righteous will be done”, could be an error for “thy righteousness will be done”. In this alternative expression, will acts as a modal verb rather than as a noun. And we can find some minor variation in the history of the text where righteous and righteousness have been mixed up, namely, in 2 Nephi 9:41, where an original “his paths are righteousness” was replaced in the 1837 edition by “his paths are righteous”. As explained under that passage, the earlier reading, righteousness, is very likely correct. As far as the proposed expression here in Ether 12:29 is concerned (“thy righteousness will be done”), usage elsewhere in the text suggests that the modal verb shall would be preferred over will in referring to something being done since there are 12 instances with shall but none with will:
In theory either reading (with righteous or righteousness) will work here in Ether 12:29. The critical text will therefore accept the consistent reading of all the textual sources rather than the proposed emendation. In fact, based on spacing between extant fragments in 𝓞, the text seems to have read righteous in 𝓞 rather than righteousness.
Summary: Maintain in Ether 12:29 the reading of all the textual sources, “thy righteous will be done”; in this phrase the adjective righteous modifies the noun will nonrestrictively.