Question: All the hedge knights that I can think of introduced in the series so far - Ser Illifer the Penniless, Ser Duncan the Tall, Ser Pate of the Blue Fork, Ser Shadrich of the Shady Glen, etc. - don't have last names, at least that we know of. The one exception I recall is Ser Creighton Longbough, but there is no mention of a House Longbough on AWOIAF, so this may be a name Ser Creighton has invented, not a House name.

I can't find any information that specifically precludes hedge knights from having House names. But (minor spoilers AFFC)

> 
>  "Knights they are," said Petyr. "Their gallantry has yet to be demonstrated, but we may hope. Allow me to present Ser Byron, Ser Morgarth, and Ser Shadrich. Sers, the Lady Alayne, my natural and very clever daughter ... with whom I must needs confer, if you will be so good as to excuse us." The three knights bowed and withdrew, though the tall one with the blond hair kissed her hand before taking his leave. "Hedge knights?" said Alayne, when the door had closed.
> 
> 
> 

Which seems to suggest that hedge knights as a rule don't have House names.

So my question is twofold:

* Is there a reason that none of the hedge knights we've met appear to have House names (Ser Creighton possibly notwithstanding)?
* Could someone from a noble House become a hedge knight and retain their House name?

Answer: 
Hedge knights are poor, unsworn to any lord or master, and go from one place to another looking for any sort of employment. It's not exactly the life for a noble. Most nobles who become knights have either trained their entire lives in castles, and can therefore find gainful employment, or can always go serve a family member. Even most bastard knights we see aren't hedge knights.

So Sansa probably made a fair assumption in assuming that when Littlefinger names the three knights, none of whom have last names, they're probably hedge knights. But she could have just as easily made that assumption if their clothes, given names, or behavior marked them as common.

There is one more example of a hedge knight with a last name: Ser Hyle Hunt, who journeys with Brienne in *A Feast for Crows*. So it's not a set rule by any means.
We know why Ser Duncan the Tall doesn't have a last name because his background is described in the short story [The Hedge Knight](http://en.wikipedia.org/wiki/Tales_of_Dunk_and_Egg#The_Hedge_Knight) (I'm assuming it's the same person). Actually, having just searched for a link to the story I see there are two sequels that I haven't read - maybe there is more in them.

Anyhow, Ser Duncan was found as a small boy living wild and sort of adopted by another hedge knight (I don't think we ever learn his mentor's name). He made up his name when asked what it was on the grounds his mentor had called him Duncan and he was tall - nothing if not logical.

The implication is that anyone can be a hedge knight just by behaving as one. Given what a miserable life it is (the story states hedge knights have been known to become robbers during lean times) the role isn't likely to appeal to anyone with enough status to have a last name.