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.
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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.
I don't think we know why most of the hedge knights we have seen don't have any last names, but a few do:

* [Hyle Hunt](http://awoiaf.westeros.org/index.php/Hyle_Hunt)
* [Creighton Longbough](http://awoiaf.westeros.org/index.php/Creighton_Longbough)
* [Clayton Suggs](http://awoiaf.westeros.org/index.php/Clayton_Suggs)

Hyle is from a noble house (the [Hunt](http://awoiaf.westeros.org/index.php/House_Hunt)) from the Reach. 

That kind of answers your second question. A person from a noble family *can* be a hedge knight.

The other two are simple/normal people that have become hedge knights.