Coleridge uses these archaic spellings in order to give the poem a more ancient feel. The mariner is supposed to live forever and every now and then detain someone whom he feels will benefit from his tale, like the wedding guest. Making the poem seem older than it is by using these archaic spellings helps to give credence to the mariner's story. The title is basically telling us the Ancient mariner has a story to tell and this is it. The Rime (or story/poem/song) of the Ancient Mariner is as follows, as the wedding guest perhaps retold it.
Rime is an archaic (old, outdated) spelling of rhyme (“poem”). When first published in Lyrical Ballads in 1798, the poem’s original title was Rime of the Ancyent Marinere. Coleridge used many archaic words and spellings throughout the poem, including rime. In later versions of Lyrical Ballads, many of these archaic words in Rime of the Ancient Mariner were edited out, and the marginal glosses (brief explanations) were added.
Some Coleridge scholars believe that rime is also a play on the word’s other meaning: “frost.” They believe that the poem is based, in part, on the second voyage of British explorer James Cook, who ventured into the Arctic Circle in the 1770s.