Here’s a bar called “Mike’s Port Pub”. (There’s only one Mike.) To me, that is a possessive apostrophe. I can see it shortened to “Mike’s” without a problem. Is this different because of the singular or first name?
If Mike’s last name were Brown, would you insist that “Brown’s Port Pub” is incorrect? If not, then “Brown’s” as a standalone name due to the understood “Port Pub” would be OK.
In that case, there is only one Brown involved, so it’s quite correct.
It occurred to me that when dealing with the entire Brown family in their more-than-one glory, you can say “The Browns” (no possessive, just a plural) or "The Browns’ " (possessive, plural, and short for “The Browns’ Place”. But you couldn’t say “The Brown’s [Place]”.
Because Brown’s would then be ambiguous. Which Brown is it?
And thank you for using Brown in your example. “Jones’s” vs "Joneses’ " would be too much to bear.
I feel compelled to provide all of us grammar nuts with a hilarious link Apostrophe Abuse which illustrates just this horrifying faux pas.
Tell me this: When discussing your friends Mr. and Mrs. Brown and all the little Brownies, would you say, “Those are my friends, the Brown”?
I hope you wouldn’t. One Brown, two Browns.
In your example of Mike’s Port Pub: Yes, Mike’s would be correct. So would Brown’s, if Mike Brown is the owner. Both are singular. But if the Port Pub were owned by the Brown family, it would properly be styled “Browns’ Port Pub.”
This is pretty basic grammar. I can’t even think of a style where “Brown’s” for a plural possessive would be correct.
Either, any or both.
I considered “Greens”, but that would have been too pedantic. 
I agree, but then we are back to the OP, where the filling-in of assumed content determines the writer’s intentions and therefore the correctness.
If my name were Brown, and I had a house sign, it would probably say, to avoid all complications, “Smith”.
And because I am as much of an apostrophe policeman as many others in this thread, even though I might be able to justify its use, I would probably leave it off the sign as the best compromise. I think what we have to remember here is a simple house sign, because of its brevity, isn’t a literary work where grammar and punctuation can be more rigidly defined.
I engrave doorknockers for a living. No apostrophe. I have told customers who insist on the apostrophe that I refuse to engrave bad grammar, and that if they are giving it as a gift I absolutley refuse. If it is for themselves, and they sign off that they insisted on the apostrophe,then I will do it…but I snigger after they leave.
No apostrophe, but further than that, no ‘S’. Just “Brown” is good enough, I reckon, implying Home of the Brown family.
If it was possessive, it would be either “The Browns’” (there is more than one Brown) or “Brown’s” (where there may or may not be more than one Brown). So, either “The Browns” or “Brown’s”. Never “The Brown’s”
That’s incorrect. The possesive plural of Brown is Browns’
Why the s?
Brown
Think of the ink you would save!
Tris
Or you could go “Sad Geek”:
Elvenhame
Kashyyyk
Mega City One
The Wookie home planet is slightly less geeky than debating whether “The Browns” is supposed to be possessive or not.
Another no. No apostrophe. No “s”.
I <heart> you.
You are a hero. I feel so happily refreshed after reading this.
Of course it can. Why shouldn’t a simple house sign be correct? It doesn’t take complex analysis to conclude that “The Brown’s” is almost always incorrect.
What if your last name ends in an “s”, like “Baggins”?
“The Bagginses”? :rolleyes:
(Yes, this is a serious question.)
Not to mention that it truly does have larger implications. If I went to an open house and the neighboring house was proudly labeled with an incorrect sign, I would conclude that the people were idiots. It wouldn’t stop me from putting a bid on my dream home, but I would think, do I want to live next door to idiots? I have been known to refuse to patronize businesses that have incorrect apostrophes on their signs or websites.
Yes, I am perhaps a grammar snob, but it saddens me when more people just don’t care about this.
Grammar snobs. Just one more reason why the SDMB is so great. 
Yes, Bagginses it is (for the plural), so put away that rolleyes. Possessive plural would be “the Bagginses’ house.”