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 :)