Do me a solid is urban slang. It is not appropriate for professional work situations.

Do me a favour/favor is informal and normal for use between friends and colleagues. 

After researching this expression, I have come to the conclusion that there appears to be no specific, generally accepted origin.

There are two speculative origins, one of which is as George has written, a subtle difference between the two, with favour suggesting reciprocity (do me a favour, I’ll do one for you) and solid conveying a commitment without expectation of payback.
The second refers to drugs (illegal I presume) in a solid form.

Because of the ambiguity of the meaning of the expression, I have tended not to use it and would suggest that it not be used except in situations where the meaning is as described by George and is both widely known and in general use.

I have never been asked to do a solid in either my private or working life.

Maybe I'm behind the times, but I have never heard anyone say 'Do me a solid'.  It sounds very odd to me -  if you said it, I wouldn't understand what you meant. It isn't standard English.

Take Dan Smith's advice and keep this expression in your passive vocabulary.  Don't use it yourself, or you'll risk confusing people if you use it in the wrong company.

"Do me a favor" is standard English. It is neither formal nor informal. It has been used for centuries. It's used in formal speech, informal speech, literary writing, and casual writing. It's "in the dictionary."

"Do me a solid" is recent slang. It's so recent that "solid," in the meaning of "favor," doesn't seem to be in the American Heritage Dictionary on line. I don't think I've personally heard it used in real life. I don't use it.

Make it part of your passive vocabulary, understand it when you hear it, but say "Do me a favor."

Do me a solid    must be slang and I've never heard it.