Although the poem was written in 1900, literally at the very end of the 19th century and therefore at the century's deathbed, Hardy probably thought better of such a depressing and relatively unpoetic title.
For though the poem starts out with a bleak and depressing tone:
I leant upon a coppice gate
When Frost was spectre-grey,
And Winter's dregs made desolate
The weakening eye of day.
The tangled bine-stems scored the sky
Like strings of broken lyres,
And all mankind that haunted nigh
Had sought their household fires.
and then continues on equally dark and gloomy, by the third stanza the sound of the thrush is heard that rescues the day (and maybe the century).
Hardy probably thought better of the name of his poem, because, although it doesn't start that way, it ends with hope for better things, at least better than a whole century on its deathbed.
Coincidentally, Robert Frost expressed a very similar idea in his poem "Dust of Snow":
The way a crow
Shook down on me
The dust of snow
From a hemlock tree
Has given my heart
A change of mood
And saved some part
Of a day I had rued.
In both poems, a bird redeems all. Wow.
Originally the poem had both titles, "By the Century's Deathbed" was a subtitle. The poem was written as a metaphor and contains a message for the readers about hope for the coming Century, a change of era, the end of a bad time, but also the end of Hardy's own life and his feelings of depair. He had recently stopped writing prose as a result of scathing criticism of "Jude the Obscure" (now one of the 19th centuries best loved books), and had well documented problems with despair, he felt that his life was drawing to an end, and his use of the word "darkling" refers to John Keats' "Ode to a Nightingale" written 80 odd years previously, in which Keats waxes lyrical about the transportative powers of birdsong.
"Darkling I listen; and, for many a time/ I have been half in love with easeful death".
He is quite possibly using the word consciously as a direct reference to these lines in Keats' similarly themed poem. This points very strongly to Hardy's feeling that he was at the end of his life, the thrush's otherwordliness bringing the hope that Hardy's own life - full of despair, may be shortly to end, and he too is "half in love with easeful death".
He actually lived another 28 years well into the next century, and enough time for the original subtitle to have lost some of it's relevance.