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: 
The only people in Westeros who have last names are the nobles.
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The rest of the populace go by a single name: Pate, Wat, Will ... etc. Sometimes with a qualifier to differentiate people who have the same name *and* live in the same place, like the villagers recruited in *Sworn Sword* (Wet Wat and Barleycorn Wat). It is only a nobleman, that belongs to a noble house, who can have a *proper* last name denoting his lineage. So Robb of house Stark is Robb Stark.

Now knights are in an interesting place. They have an honorific placed in front of their names, are a step above the commoners, but that honorific doesn't come with anything else. No lands, thus no noble house, thus no last name. So unless a knight is already of noble birth, he won't get a last name just by being knighted. 

Hedge knights, by definition, are poor knights who are not sworn to any single lord on a permanent basis. The vast majority of them are of common stock, since nobly born knights tend to find employment easier with other nobles, which gives us a lot of knights living in the hedges with no last name. To give themselves proper *airs* most of those titleless knights adopt a qualifier of some sort to distinguish them. A home village (Ser Arlan of Pennytree), or a physical trait (Ser Duncan the Tall), and so on.
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.