Do me a solid meaning "do me a favor" (ODO) is an AmE slang expression whose origin, despite its recent usage, remains unclear. The issue is why "solid" came to mean "favour". Among different hypotheses (solid meaning drugs, for instance) the "truncation" one appears the most interesting. Do me a solid "favour" is also predated by expressions like, "do me a solid "service". Earliest usages date back to the '60s, but the expression was made popular by a TV show in the 1990s. The following extract has the story:

• Semantic/pragmatic note: some commenters on the net treat solid simply as a stylistic variant of favor — and a few consequently deride it as needless slang — but some people see a subtle difference between the two, with favor suggesting reciprocity (do me a favor, I’ll do one for you) and solid conveying a commitment without expectation of payback.

• Historical note: The use is specifically American and is associated with the young and hip. New Partridge has cites back to 1973 (in Connie Eble’s compilation of UNC college slang). Some sources suggest that the origin of solid ‘favor’ lies in drug use (from solid ‘drug in resin form’, presumably truncated from solid dope or solid hash). Here’s one potential example of solid in a drug context:

• “Say, you like sick, like you need a fix / Perhaps I can do some solids for you.” (Dennis Wepman et al., The Life, 1976, cited in New Partridge)

• New Partridge has this under solid ‘favour’ but it seems more likely to be a drug use. The semantic development from ‘solid dope’ to ‘favor’ is hard to work out, and solid could easily arise as a nouning by truncation independently in different contexts: from solid N (N = dope, hash, etc.) in a drug context, from something like solid favor in other contexts — and, indeed, from solid pipe in still other contexts and from solid N (N = dope ‘information’, information, etc.).

• My proposal is that solid ‘favor’ is a nouning by truncation, around in U.S. slang for about 40 years (at least). But it’s surged in the last 20 years, almost surely through the influence of the tv show Seinfeld. From “The Jacket”, first broadcast 2/6/91:

  • (Enter Kramer) KRAMER: Hey. Hey, would you do me a solid?
    JERRY: Well, what kind of solid?
    KRAMER: I need you to sit in the car for two minutes while it’s double-parked. I gotta pick up some birds.
    JERRY: Birds? […]
    JERRY: I can’t. I’m on my way out. There’s no way I can do it.
    KRAMER: George, do me a solid? Two minutes.
    GEORGE: Well, I’m going with him. I’d like to, I’ve never done a solid before.
    KRAMER: Alright.. yeah.. alright, have a good one. (Kramer leaves)

(arnoldzwicky.org)