The earliest text reads “great desires”, although the singular desire is possible. The 1858 Wright edition accidentally set the singular, but this reading did not persist. Elsewhere, there is one other example of “great desire(s)”, and it too reads in the plural: “because of their great desires which they had for the welfare of this people” (Alma 60:9).
Summary: Based on the earliest textual sources, the plural “great desires” should be maintained in 1 Nephi 2:16.