Nobody now alive has seen a great country, so what they mean is some sort of wishful thinking.
We have an entire nation of people who can talk all day long without knowing what exactly they are saying. So it's kinda pointless to discuss what they mean.
Economists (actual economists, not the government bullsnot artists) have been predicting collapse since 1964 when silver backing of the money was abolished, and since 1971 when gold backing was abolished. The only thing that has changed is the verbs are in present tense now, instead of future tense.
Economists are able to predict a collapse because symptoms are quite obvious and no country has ever avoided collapse after the symptoms began to show. But timing and details are random.
I think the sense of that, as President Trump meant it, was that this country,in its institutions and understandings, was great as it was but we have allowed other special interests and political correctnesses to sidetrack us from what is truly important. Which I agree with.
I think it's simply a lack of imagination - they couldn't think of a new slogan so they just added a word to the old one. Anyway at this very moment the new Make America Great Again hats that Trump ordered from China are held up at customs - if Trump truly believed in making America great those hats would of been made in the US not China. Cheers!