The song's theme is optimism. The very title speaks to the idea that, whatever the setbacks, one's goal can be achieved with persistence.
One notable feature of the song is its lack of consistent rhyme. The chorus is a/a/b/a, with the first and last lines ending in "day." The first verse's lines end with the words "mirror/you/of/love," for a rhyme scheme of a/b/c/c. The second verse: "dreamer/thinkin'/doin'/say," or a/b/c/d (which is to say, no rhymed lines). And then the third verse is "discouraged/slowly/doin'/movin'," so the slant rhyme of the last two lines brings us back to a/b/c/c.
Perhaps this lack of rhyme reflects the state of mind of the listener-- not so much the person speaking, but the one being spoken to. The listener is someone who is "down" and needs to be "bucked up." He (or she) is not feeling like there is much rhyme is his life.
The lyrics themselves are very straightforward and comprise a "pep talk." The way The Seekers perform it, it could be addressed to anyone needing encouragement.
While I am a bit hesitant to assign meanings to Simon's songs based on his circumstances at the time of their writing, I must wonder if in this case such an ideas isn't warranted.
Simon and Garfunkel, no longer Tom and Jerry, had regrouped as a folk duo and put out "Wednesday Morning." It did not do well. So Simon might have been addressing the song to Garfunkel, telling him that with some persistence, they might still find success.
Another, even bolder, interpretation might be that Simon addressed this song to himself. In England at the time, he recorded an album of solo acoustic material (his "Songbook") and was writing songs with Woodley, but neither was the success he had hoped for with Garfunkel.
One can imagine him "look(ing) in the mirror," trying to convince himself that in a "time not so far away," that could still happen.
What he did not know is that, back in New York, a producer named Tom Wilson was creating an electric "remix" of "Sound of Silence"... which would go to #1 and start Simon and Garfunkel on their way.
Another collaboration with Woodley, this became the song that put The Seekers on the map, hitting #11 in the UK and #4 in Australia.
As The Seekers were the first act with international impact from Australia, it is fair to say that Simon, aside from his part in South Africa's musical history, is also part of that island continent's cultural story.
Later acts from Australia ranged from The Little River Band to Men At Work, and perhaps most notably another harmony act like The Seekers-- The Bee Gees. Well, not much like The Seekers.
Next song: My Little Town