The poem begins and ends with the image of "frost at midnight" as it performs its "secret ministry"—freezing the earth and water, and forming icicles (1-2, 71-75). "Ministry" naturally makes us think about Christianity—the frost is somehow like a pastor. But Coleridge is also playing on another meaning of the word: a "minister" is someone who does something on behalf of someone else (a closer word in modern parlance would probably be "agent"). So, the frost is like an agent of God, working to do… well, something (make icicles? reflect light from the icicles back to the moon?) on God's behalf.
Coleridge doesn't want us to see Nature—and everything that is a part of Nature (like the frost)—as something dead and mechanical. It's not, in his view, just a bunch of physics equations and atoms (though it is that too). Coleridge wants us to see Nature as pulsing with life-force, as a collection of powers and processes that all testify to the supreme power of God. That's why he describes Nature as the "eternal language" (61) of God. It gives testimony to his existence, in Coleridge's view.
In the poem, Coleridge is basically praying that his baby son—Hartley Coleridge—will be able to take joy and comfort in Nature, no matter what the season. The icicles created by the frost will still bring evidence of God to humanity, in the same way that they reflect light from the moon. "Frost" could easily be a symbol of something bad—paralysis, the destruction of crops and living things, even death itself. But, instead, Coleridge makes it—and all of Nature—a symbol of something that is basically good. This isn't just how he wants his son to live with Nature. It's how he wants his readers to live with Nature, as well.