Here the 1841 edition replaced the archaic use of mine before a vowel-initial word with my, the form that modern-day English speakers expect. A similar change occurred in the 1841 edition for two other cases involving own, and the second of these has persisted in the LDS text (just like here in Omni 1:10):
There are a few other instances where the 1841 edition replaced mine with my in attributive position (that is, where it modifies a following noun):
In a couple of instances, the early RLDS editions have also shown this tendency to replace mine with my:
But the vast majority of cases of mine in attributive position have remained (130 occurrences). Thus the sporadic changes in the 1841 edition do not seem to reflect any systematic attempt at editing but rather an accidental tendency for the compositor to set what modern English speakers expect before nouns.
When own follows mine or my, the Book of Mormon text prefers mine (47 to 7 in the original text). The critical text will follow the earliest textual sources in determining whether the reading is mine or my before vowel-initial and h-initial words. For a complete discussion, see possessive pronouns in volume 3.
Summary: Restore in Omni 1:10 (and Ether 3:24) the original reading mine own.