The title isn't a title so much as it's the first line of the poem, a trait typical in Dickinson's work. She rarely gave her poems separate titles that would offer a reflective statement on the work, so calling it "Precious Words" is more a later editorial choice.
Within that first line, however, we can see that the wording is significant: "He ate and drank the precious Words--." This is a poem that points to Dickinson's familiar theme about the power of poetry but uses the conceit of freedom to do so. Eating and drinking precious words liberates the "he" of the poem, giving him wings to rise above the universal problems of poverty and mortality (Dust).
However, this poem has elements that recall other poems by Dickinson that flirt with questions of spirituality or religion. The poem suggests more than just an escape from reality through books. It seems to offer almost a metaphysical transcendence, not unlike other Dickinson poems ("I reckon when I count at all"). That prompts one to reflect more on what the precious Words might be and how one eats and drinks them. In Christian practice, Christ is referred to as "the Word." The Gospel of John begins with the line "In the beginning was the Word, and the Word was with God, and the Word was God." This divine Logos tempted poets from at least the Renaissance to play with the idea of language and divinity, of poetry and god-like creation. This reference would make sense of eating and drinking "the precious Word," for that is what Christian communion requires: to imbibe the Logos of God and to receive salvation in doing so.