How can a US citizen sponsor an illegal immigrant from Mexico so that they may remain or return to the US?

Your question reveals an unconscious bias. You said,

= "How can a US citizen sponsor an illegal immigrant...." = If you "sponsor" them, they are not an "illegal immigrant." Careful, the way your question is phrased says a lot about you. The very term "illegal immigrant" is insulting to ....migrant workers, especially. Mexican citizens surely don't like the phrase and, by the way, Superman was apparently an "illegal immigrant" and you SURELY wouldn't want to insult HIM.