It's notable that it rhymes on solidarity, which offers precisely the semantics that are needed to explain "do me a solid".
From wiktionary (https://en.wiktionary.org/wiki/solidarity):
A long time union member himself, Phil showed solidarity with the picketing grocery store workers by shopping at a competing, unionized store.
Only the solidarity provided by her siblings allowed Margaret to cope with her mother's harrowing death
Why this should have been clipped is easy to see. The adjective solidary is a bit of a moutful, and allusion to solid = strong, real makes it a neat wordplay, albeit demoting it to a mere emphatic.
Update: search for "solidary favor[u]r" yield absolutely no results, but also show that the word is an odd-ball in English, often confused with solitary, and google suggesting "solidarity" instead, yielding a few insignificant results for "solidarity favor" (often with favors as a verb) and "Solidarity's favor" (which in many cases seems to be using the proper name of a political entity in Australia?).
Anyhow, a look at the etymology section over at wiktionary reveals that solidarity derives through French from Latin solidum, which indeed meant solid. So the admitted allusion seems to be more than mere coincidence.