Lann the Clever founded House Lannister, and King Durran Godsgrief founded House Durrandon (ancestors of the Baratheons,) but would you name them as "Lann Lannister" or "Durran Durrandon"? It seems strange to me, and I can't find anyone else who does it. Yet they founded those respective houses. So the question is: should you apply the house name to their founders? Or should they stay Lann the Clever, and Durran Godsgrief?
tl;dr: It's hard to know for sure, since all of the information we have from that time comes from legends and stories passed down by word of mouth(+), but I would say that it's highly unlikely that those two individuals would have ever been called "Lann Lannister" or "Durran Durrandon". Most likely, their House names did not become commonplace until long after those "founders" had died.
For starters, it's important to note that the type of noble houses that exists in Westeros in the time of the novels is likely much different from the ones founded by legendary figures like Lann the Clever, or Durran Godsgrief.
The current form of nobility exists, in essence, at the pleasure of the King, and only the King can create new houses. In modern-day Westeros (like most of modern-day Earth), it's typical for even non-nobility to have a family name, and that would become the name of their new house. For example, Aegon I Targaryen created House Baratheon for his (rumored) half-brother Orys Baratheon, and Joffrey I Baratheon created House Slynt for Janos Slynt.
The houses you're talking about, on the other hand, like Stark and Arryn and Lannister, date back thousands of years, to the Age of Heroes, which leads to a few problems trying to figure out how they would have been named.
The first of those problems is deciding if they are even real people. According to Martin, these men are not supposed to be historical figures, but legendary archetypes of the "old days". When talking about Bran the Builder, building the Wall, Martin says:
No one can even say for certain if Brandon the Builder ever lived. He is as remote from the time of the novels as Noah and Gilgamesh are from our own time.
Even in-universe it's not clear how much of those stories are true, as Sam says:
There are archmaesters at the Citadel who question all of it.
Even if those people existed, we have no real records of how they came to "found" their houses, nor any idea of what the naming conventions in effect at the time were. One thing we never see, however, is any last name assigned to the prominent figures of that time. They are always identified by their sobriquet, like "Bran the Builder" or "Lann the Clever" (in this sense, "Godsgrief" was not Durran's last name, but rather a sobriquet.) We certainly never hear them identified as Bran Stark(++) or Lann Lannister, which we can speculate means that the naming conventions at that time were probably much more primitive, and that those people probably pre-dated formal surnames.
We also have no concrete evidence to prove that those specific individuals actually founded the houses that claim to be descended from them. The concept of noble houses, IMO, seems out of place for that time period. What I think is probably more likely (though again purely speculative) is that current Great Houses were not literally founded by those legendary figures, but rather, by their descendants, using their ancestry to bolster support for themselves.
As an example, House Lannister claims to be descended from Lann the Clever "through the female line". That seems odd if, in fact, Lann himself has founded the house: they would simply claim direct descent from Lann himself. However, if we assume that Lann existed, but did not actually found his own house, we can speculate that Lann's descendants began calling themselves Lannister, in the same sense that many early real-world surnames were originally patronyms (the afore-mentioned Erickson, or the Gaelic MacDonald or O'Connell). At some point down the line, someone important married one of the Lannister women, and decided to take her name, and thus her connection to Legendary Lann the Clever, when forming his own House.
A similar chain of events would have led to the eventual formal founding of the other noble houses, some of which had patronymic surnames (Durrandon) and others did not (Stark, Arryn). Either way, the people usually identified as the "founder" of those houses would not have used those surnames for themselves.
(+) There is, apparently, a lot more backstory from that time period available in the now-defunct A Song of Ice and Fire role-playing game material, but I have never read any of that, and wouldn't consider it to be canonical, since ASAIK Martin had nothing to do with it.
(++) UPDATE: There is, apparently, a DVD extra on the Season 1 Box Set, called a "Complete Guide To Westeros", that does identify Bran the Builder explicitly as "King Brandon Stark". However, none of this content was written by Martin (rather, by a staff writer for the show), and since this is the only time that he's ever been given a surname, my instinct is to call this non-canon and ignore it. YMMV.
No, the founder would not share in the house name as far as titles go anyway.
The "ister" suffix is from the same vein as "cester" which is a Saxon suffix given to places that were forts or encampments, specifically Roman ones. So, we have Lann-cester or Lannister to make it flow. It seems my information regarding them being places of commerce was false.
"Don" is a suffix from "dun" meaning "hill". Durran Hill/Mountain/Highlands etc. It makes little sense to complicate Durran's name to "Durran of Durran Hills".
Further, the purpose of the house name is to associate Durran with those of Durrandon, it's redundant to associate Durran with Durran.
I think that this is more like a nordic type of naming.
In Sweden there is the suffix -kvist which means twig. In other languages (like German) 'a family branch' is translated to 'a family twig'.
So you'd have a branch of the family of Lann. Same with Durran and all the others named in this manner.