The Book of Mormon text basically uses the two verbs administer and minister interchangeably. In some instances, modern speakers of English prefer minister over administer (as here in Alma 15:18); in some other instances (discussed below), administer is preferred over minister in modern English. Nonetheless, there has been relatively little textual variation between these two verbs in the Book of Mormon text. Here in Alma 15:18, the 1874 RLDS edition replaced administer with the expected minister (but perhaps unintentionally since that change was never made elsewhere in the 1874 edition).
In the original Book of Mormon text, there are 56 occurrences of the verb minister and 29 of administer. There are only two other cases that show textual variation:
The first case shows a strong tendency to replace administer with minister: (1) Oliver Cowdery initially wrote minister in 𝓟, then virtually immediately corrected minister to administer (the probable reading in 𝓞) by supralinearly inserting the prefixal ad; (2) John Gilbert, while preparing the manuscript for typesetting the 1830 edition, used a pencil to cross out Oliver’s inserted ad; yet when it came to actually setting the type for the 1830 edition, the word was set as administer; and (3) the 1858 Wright edition introduced minister into the RLDS textual tradition.
A third case of variation between minister and administer may have occurred in the original manuscript for Alma 22:25. The earliest extant text (the printer’s manuscript) here reads “but the king stood forth among them and administered unto them”. But spacing between extant fragments of 𝓞 suggests that Oliver Cowdery initially wrote ministered in 𝓞, but then he immediately crossed out the whole word and wrote inline the correct, but unexpected, administered (for further discussion, see under Alma 22:25). This use of “administer unto someone” instead of the expected “minister unto someone” is found three other times in the text (in addition to the four examples in Alma 15:18, Alma 17:18, and Alma 22:25, cited above):
These three examples have avoided the tendency to replace administer with minister, the verb that we expect in modern English for this context. But the other examples show the persistent tendency to replace “administer unto someone” with “minister unto someone” in the Book of Mormon text. Overall, the original text has examples of both the unexpected “administer unto someone” (7 times) and the expected “minister unto someone” (42 times).
On the other hand, in modern English we expect “administer the sacrament”, not “minister the sacrament”. But in this context, the Book of Mormon text is basically divided in its usage, with three examples of minister and four of administer:
As noted above, the single instance of minister in Moroni 4:1 was replaced by administer in the 1841 British edition (but in only that edition).
According to the Oxford English Dictionary, administer is historically a 14th-century Latinized re-formation on the French word minister, so it is not surprising that the Book of Mormon often uses both words in the same semantic context. Over time, English speakers have differentiated the senses of these two words so that administer and minister now generally occur in nonoverlapping contexts (thus, a judge administers an oath, a reverend ministers to his congregation, the elders administer to the sick by the laying on of hands, and the priests administer the sacrament). But the Book of Mormon shows the earlier variability in the language between administer and minister. For each case of (ad)minister, the critical text will therefore follow the earliest textual sources in determining which verb, administer or minister, is intended, thus allowing for textual variation that sometimes goes against what we expect in modern English.
Summary: Accept in Alma 15:18 the original use of administer rather than minister (“and did administer unto him in his tribulations”); the original text of the Book of Mormon frequently uses the verbs minister and administer interchangeably in contexts where modern readers expect only one of the verbs.