Why did Dylan Thomas choose the title "Do Not Go Gentle into That Good Night"?

This is an interesting question if properly examined because, while the simple answer is that actually Thomas didn't give his work this title, this leads us on to the broader topic of the lack of titles in Thomas's poetry as a whole. Put simply, Thomas didn't seem to be a fan of putting titles on his poetry—or even on his collections of poetry. His collection 18 Poems, for example, gives no indication on the front cover of what the reader can expect inside, and all of the poems in it are similarly without titles. They are known by their first lines, as was historically the case with a lot of poetry. Think about Shakespeare's sonnets, for example—they are numbered, but not named, and are generally known by their first lines.
So, why didn't Thomas want to use titles? What does it do to us as readers when there's a title on a poem? Generally, it can give us a guide to what the poet thinks we should be thinking about—that is, what their main thought was when writing the poem, or what the poem is supposed to evoke. By not titling his poems, then, Thomas allows us to think more freely about his poetry. Instead of being told, or guided toward, a theme or focus in the title, we have to experience the poem entirely on our own. Its key focus is up to us.

This poem is actually untitled and has become known as "Do not go gentle into that good night," its first line, as a result. You can tell because, in professional publications of the poem, only the first letter of the first word is capitalized instead of each significant word being capitalized. It is an appropriate title, however, because it encapsulates one of the main purposes of the poem: it comprises an attempt to persuade the speaker's father to fight against death and not go quietly into its symbolic darkness. In line 16, the narrator refers to his father as being on "that sad height," a figurative precipice that we can understand to signify a choice between moving forward into the darkness or stepping backward into the light. The speaker urges his father to "rage against the dying of the light" and to resist any desire to slip away. The line used as the poem's title does likewise.