I first learned drop on by from this news.
Longhorn Aquatics was hosting its New Year's Classic, and Lance Armstrong decided to drop on by and hop in the pool.
What is the difference between drop on by and drop by? What is the meaning of on here?
Since no one else came up with a thoroughly researched reply...
I suspect that we get it by evolution from "come on over". The "on" means "forward" or "ahead" literally, but in the phrase it adds encouragement, like you'd use with a dog for example.
In addition to the "encouragement" aspect @JeffSahol mentions, "drop on by" is slightly more informal than "drop by" (which is itself already fairly informal).