The Invitational On
In the imperative mood, supplementing a phrasal verb with the particle on in familiar, colloquial speech turns the usual command/request into more of an invitation:
“Come on by the house. I got something for you.” — Susan Wilson, The Dog Who Saved Me, 2015.
If there is a declarative sentence “*s.o. came on by the house,” it was not uttered within earshot of the Google search engine. The particle on here only has the purpose of blunting the imperative.
“Come on over,” which sounded almost the same, and so amiable and smooth, was a way of inviting someone to your house. — Lynne Sharon Schwartz, Leaving Brooklyn, 2011, 68.
Are you cringing at the thought of a white Christmas? Are you dreading the idea of sludgy sidewalks and sub-zero temperatures? Suffer no longer! Just fly on down to sunny South Florida and see how sweet the winter months could be. — Website, Bocaire Country Club, Boca Raton FL.
So, whenever you get finished with your banquet, drop on by. I'll have ’em set up a tab for you. — Old Boyfriends (filmscript), 1979. COHA
Invitational on is a common feature of Southern American speech, including AAVE, which accounts for its use in lyrics and titles in countless popular songs from Rufus-Chaka Khan’s “Stop On By” to “Drop On By” by Laura Bell Bundy (b., Lexington KY). This usage in the imperative, while culturally visible, is not exclusive to the South:
Twice the beer, twice the joy. Drop on by and grab our beers at their current promotional prices. — Mad Bistro Whisky Lounge, Singapore.
Drop on by for Saturday fun! — Headline, Morse Messages, Morse Institute Library Newsletter 2, 4 (Nov. 2016), Natick MA.
Come on by on Saturday morning and kick the weekend off with us! — Kung Fu studio, London UK.
So if you fancy taking TiVo for a test drive or finding out how you can get broadband in as little as 5 hours, come on by and our friendly store team will happily get you sorted. — Virgin Media Store, Doncaster UK
If you live in the area, please come on by to say g’day! — Air purifier manufacturer, Sydney, Brisbane, Melbourne, Australia.
Stop on by. We’d love to show you around. — Event venue, Walla Walla WA.
Unlike come by, drop by does have the non-imperative variant drop on by. Contrasting drop on by with drop by and a non-existent *drop on, linguist Bert Capelle calls the particle on “an embellishment.” It would be difficult to argue with this analysis were it not for the difference in register, which your example sentence aims to evoke: with drop by already casually colloquial in every variety of English, drop on by is ignored by dictionaries because it adds a nuance of friendly familiarity beyond a mere dropping by somewhere, and such nuances are impossible to lexicalize.
This nuance is especially transparent in rhetorical contexts that are not necessarily by nature friendly or familiar:
Jenny Greenteeth has dropped on by over at the Flames Rising website. Drop on by and have a swim — if you dare. — Evil Hat Productions.
Both on by and over at are designed to maintain a highly informal, convivial atmosphere on a website run by a small crew of game designers. The linked article stands in as a metonymy for the author, who is depicted as if she just happened to be in the neghborhood, i.e., a physical space, when she happened to drop by over at another website and wrote an article. This, of course, is a metaphorical fiction, but it serves a rhetorical purpose.
The same fiction is maintained by various bloggers and podcasters:
Shanghai Kate dropped on by the studio today and as always it was an honor for her to come through. Shanghai Kate is known to many as America’s Tattoo Godmother. — Skin Design Tattoo (Las Vegas), 9 April 2015.
This month, Palmbomen dropped on by and shared some incredible sonic selections with all of you. — Mixcloud.com
This week Tegan Higginbotham dropped on by! We chatted Sovereign Hill, Girls toilets, Bikram Yoga, Drunk on bread? and Maccas Delivery. — Nath Valvo Podcast (Australia), 19 July 2014.
Unlike the game design site, these are human encounters in three dimensions with spoken language the listener may hear, but the notion that any of these visits were either unplanned or unannounced is absurd. It’s all to set a tone that includes the listener as a familiar friend.
There are two aspects of dropping by somewhere. A person might be out and about and suddenly decide to go to a particular place, or if the object is to visit a particular person, the visit may be unannounced, but the relationship is such a familiar one, it would not be socially awkward. In authentic language, i.e. when a person’s use of a particular register is natural and unreflected, free of any obvious rhetorical strategy to simulate such a register with all its warm connotations, Capelle’s “embellishment” functions more as an intensifier.
Was with kids in that part of town they were hungry it was post breakfast and nearing end of lunch so we dropped on by… — Trip Advisor review, Tearoom, Falkirk (Scotland).
My partner and I were feeling a bit peckish after our movie and so we decided to drop on by the Yates Street Tap House for a light snack treat and beverage. — Trip Advisor review, Victoria BC.
The on particle accentuates the spontaneity of the decision to go to a particular restaurant, but could just as easily be left out. There is also no way of knowing if the expression is favored by these speakers over the “unembellished” variant.
To make it even more unbelievable I had the entire beach to myself. Not a single soul around for the first couple hours. A boy who lived nearby saw us and dropped on by to have a chat for a short period of time and then low [sic] and behold, the Slovenians turned up again. — Travel blog, native Alaskan.
Wherever the boy is planning to go, he sees the woman on the beach and heads over for a brief conversation. Again, the on intensifies the spontaniety.
I work for a huge state university on its flagship campus. This sometimes might work, but mostly we’re just understaffed and overworked and you would actively make enemies if you just dropped on by someone’s office (which may be located a mile or more away from your own office). — Comment, askamanager.com
Dropping (on) by someone’s office because they have not responded swiftly to an email — especially someone whom one might know only fleetingly or not at all — can “actively make enemies.” It assumes a familiarity that isn’t there.
At some point a staff member dropped on by, but was fine with us using the room to shoot in. We didn’t shoot for long since I didn’t want to press our luck and it was really stuffy in the room. It was a pretty quick shoot, but I felt that a lot of good shots were obtained in that short amount of time. — Kelly Dun, photographer, San Francisco.
Like the boy on the beach, a staff member at this location, seeing the photographer and his model doing a shoot no one knew about, is curious and stops to ask.
The difference between drop by and drop on by is primarily one of register: cultivated as a rhetorical fiction or used authentically by people from the US, Canada, Scotland, and Australia. While there may be regional differences — the writer from Brooklyn seems unfamiliar with the extended phrasal verb — the presence of the expression among speakers from all over the Anglosphere suggests a breadth of usage.
What the on particle does not do is suggest directed motion (“Get on up here”) or further iteration (keep on doing vs. keep doing). Like its near synonym stop by, which also has a non-imperative variant, dropping and stopping are punctual, like an arrival: you don’t drop or stop by until you’re already there.