First off, JKR never came out and stated either way.
So we can only infer.
I'm inferring that she didn't intend the name to make fun of the character .
First, generic evidence - most of her names that are meant to convey meanings are NOT meant to make fun of someone (except for "looney" Luna's first name, possibly) or otherwise hurtful.
Nearly all the meaningful names (Lupin, Weasley, Newt Salamander, Voldemort, Lovegood, Poppy, Ollievander, etc...) had real non-insulting meaning, either in-universe or out of universe.
So, naming Neville with a deeply insulting name intentionally would have been out of pattern.
Leaving that aside, we have evidence from the name itself: t's a real English surname, with fairly non-funny and non-insulting meaning/origins.
This interesting name is of Anglo-Saxon origin, and is a topographical surname for someone who lived in a long valley or dell .
In some cases the modern surname, found as Longbottom and Longbotham, may also be locational in origin, from the place called "Longbottom" in Luddended Foot in West Yorkshire.
In either instance the name derives from the Old English pre 7th Century "lang", Middle English "long", long, with Old English "bothm, botm", Middle English "bodme", bottom, valley, dell.
The first recorded spelling of the family name is shown to be that of Richard Longboteham, which was dated 1379, in the "Yorkshire Poll Tax Returns", during the reign of King Richard 11, known as "Richard of Bordeaux", 1377 - 1399.
Source:
http://www.surnamedb.com/Surname/Longbottom Please note that the fact that JKR intended it to be a topographical meaning of the name is even more likely considering that his first (given) name is ALSO topographical: This great and noble surname is of Norman origin, introduced into England after the Conquest of 1066.
It is a French locational name from "Neuville" in Calvados or "Neville" in Seine-Maritime, Normandy, both so called from the Old French "neu(f)" new, with "ville", a settlement (src: http://www.surnamedb.com/Surname/Neville )
So, for that matter, was another earlier version of Neville's surname (from the First Forty document referenced in @Richard's answer): ...
either a locational surname, from a place called Sidebottom in Cheshire near Stockport, or a topographical name peculiar to the counties of Cheshire, Lancashire and Yorkshire.
The derivation of the name is from the Olde English pre 7th Century "sid", wide, broad, spacious, and "bothm", valley, bottom, dell, and in the case of the topographical surname, this would denote residence in such a "wide valley" .
(src: http://www.surnamedb.com/Surname/Sidebottom )
An interesting contradiction is that, on the same First Forty document, Neville also had another rejected version of surname, Puff.
That one doesn't seem to have much of topographic origin, and isn't even British or French.
The only known origins are German/Austrian: nickname for a violent, aggressive person, from buff ‘push’, ‘shove’.
From a pet form of a personal name, Bodefrit, composed with Old High German biutan ‘to bid or order’ or boto ‘messenger’.
Possibly an altered spelling of Pfaff.
If that's the meaning, I can see where JKR was coming from crossing that name out :)
scifi.stackexchange.com/users/976/DVK-on-Ahch-To
I think we can be pretty certain that JKR's intention was to make Neville sound a bit "goofy" with her choice of name.
Longbottom certainly isn't the most dignified of names, despite it being a traditionally English surname.
Note also that HP is a children's book.
Names that contain the word 'bottom' are generally considered hilarious.
On top of that, the " first forty " document (showing her initial thoughts on character names) listed him originally as Neville Side bottom , another slightly unusual British name and one that is, to some quite amusing.
Obviously she had bottoms in mind when choosing his name.
scifi.stackexchange.com/users/20774/Valorum
It is a real name, not something JKR made up, and it has been used in England for a long time.
This interesting name is of Anglo-Saxon origin, and is a topographical surname for someone who lived in a long valley or dell .
In some cases the modern surname, found as Longbottom and Longbotham, may also be locational in origin, from the place called "Longbottom" in Luddended Foot in West Yorkshire.
In either instance the name derives from the Old English pre 7th Century "lang", Middle English "long", long, with Old English "bothm, botm", Middle English "bodme", bottom, valley, dell.
The first recorded spelling of the family name is shown to be that of Richard Longboteham, which was dated 1379, in the "Yorkshire Poll Tax Returns", during the reign of King Richard 11, known as "Richard of Bordeaux", 1377 - 1399.
Source:
http://www.surnamedb.com/Surname/Longbottom
scifi.stackexchange.com/users/1148/Jack B Nimble
My last name actually is Longbottom, just to confirm it is definitely a name which is still around today.
A lot of Neville Longbottom jokes are a burden of the name.
scifi.stackexchange.com/users/66975/Anon
Yes.
Fact: "
Longbottom" is a funny name.
Maybe not the funniest name possible ( like "Hugh Morris" for example )
but there is undeniably a little bit of humor in this particular choice of names.
And if you can remember back to being the age/maturity of the target audience of the book I think you'll remember snickering and giggling about all things "fanny", "bottom", "butt".
scifi.stackexchange.com/users/43084/David Eugene Pratt
Is the family name of Neville a funny one that is actually part of the his "image" in the first books?  I'm asking this from the point of view of a non-British and a non-English native speaker.  Another form of this question would be - is Longbottom a name that you rather not give to your children if you can avoid it.
Did Rowling give Neville the family name Longbottom as a joke/funny name as part of his initial image
scifi.stackexchange.com
<harry-potter>
scifi.stackexchange.com/questions/85880
scifi.stackexchange.com/users/40429/vap78
"Longbottom" is a funny name, and although it is  a real, traditional name, we can be pretty certain that JKR's intention was to make Neville sound a bit "goofy" with her choice of name. Although the writer’s intention is not to ridicule the character by choosing the surname Longbottom, he/she clearly wants to make the character sound a bit goofy. Moreover, Longbottom is a genuine name albeit not the most dignified. 
It is obvious that the word "bottoms" was in the mind of the writer when choosing the name of the character and her intention was to make the character sound goofy by choosing a funny name. Longbottom is a real name albeit not one of the most dignified. The author’s intention was not to ridicule the character despite Longbottom jokes being commonplace. 