It's an interesting question. The problem is that "surnames" did not become a thing until a couple of hundred years after the Norman Conquest. They caught on with the aristocratic families first as they became known by "surnames" connected to towns they owned back in Normandy. For example, the Percy family took their name from the town of Percy-en-Auge. It was different for the lower orders, however, as surnames were not adopted round the time of the Norman Conquest but over the next couple of hundred years as they were needed to identify people for taxation purposes. Consequently, people did not take their surnames from their lords but from things that would identify them in a small town setting that would differentiate you from other people who had the same first name as you. These are sometimes called "eke-names". For example, John Short might have been so named because he was short, while John Bytheway might have been called that because he lived next to the main road and John Moxon would have got his name because he was the son of a formidable woman named Margaret (Moxon being a contraction of "Meg's son"). If John Short, John Bytheway, and John Moxon all lived in Leicester and a new John arrived in town saying he was from Derby, pretty soon he would be known as John Derby. When John Short, John Bytheway, John Moxon, and John Derby died and passed their farms and small tenements onto their sons, William Short, James Bytheway, Robert Moxon, and Henry Derby would use their father's "eke-names" as surnames to establish their connection to their father's property. The poorer you were, the less property you had, the less need for a surname.
So, what influence did the Normans have on surname practices? My own surname is actually an interesting example in this case. "Emery" derives from the Germanic first name Emaurri or Amaury, which was a popular first name for conquering Normans. You would think that would imply I am descended from a Norman. However, that's not necessarily true. Cecily Clarke has argued that Anglo-Saxons utilised Norman names for their sons as a way of integrating themselves with their new overlords in an attempt to climb the social ladder, while giving their daughters Anglo-Saxon names to preserve family memory. Around 1066, it is clear to see that all the Anglo-Saxon men had names like Æthelwulf, Cynewulf, and Æthelred. However, by 1100, England is full of Roberts, Richards, and Williams. Not all of these were of Norman descent. While some were of mixed Anglo-Saxon and Norman descent (the chronicler Orderic Vitalis is a good example of this), others were fully Anglo-Saxon. It just seems that many Æthelwulfs had decided to name their sons Robert to allow them to climb the ranks more easily. This trend meant that by the fifteenth century, England was full of Williams, Matildas, Alices, and Roberts. The only Anglo-Saxon name that "makes it" through the medieval era is Edward, and that is only because the Anglo-Norman king Henry III (1207-1272) decided to name his eldest son after the Anglo-Saxon saint Edward the Confessor. The other Anglo-Saxon names that are perhaps more familiar to us as first names (eg. Edwin, Edith, Alfred) were almost entirely resurrected by the Victorians.
So, let's see how the acquiring of a "Norman" surname by an Anglo-Saxon plays out. Let's say my hundred times great-grandfather was an Anglo-Saxon peasant called Æthelred with a wife named Beorhtgifu. England is invaded and their local patch of land is taken over by a guy named Amaury, who quickly constructs the local castle. Wanting to be on Lord Amaury's good side, Æthelred and Beorhtgifu decide to name their son Amaury after their new lord. Amaury grows up on Lord Amaury's estate, and due to interactions with the Normans soldiers brought over by Lord Amaury, Amaury picks up a bit of Anglo-Norman. Through this, he gets a job at Lord Amaury's castle. Lord Amaury dies, and his castle is inherited by his son Lord John. Lord John gets married to the daughter of a local Norman lord, Alice, who has employed an Anglo-Saxon servant called Æthelflæd. Amaury and Æthelflæd fall in love, get married, and have two children, who they name John and Alice after their bosses. Lord John and Lady Alice are happy for their servants, and award them a small tract of land just outside of Leicester.
Amaury and Æthelflæd move to the farm with their children. When the taxman comes round, Amaury puts his name down as "Amaury de Leicester", as he is the only Amaury in the area. When Amaury dies, his farm is inherited by his son John. When the taxman comes round again, John has to distinguish himself from the other Johns who could also go by the name "John de Leicester" - John Short, John Bytheway, John Moxon, and John Derby - so decides to take his father's name and becomes John Amaury. John Amaury gets married to John Bytheway's sister Alice, who is of full Anglo-Saxon heritage but was also named after Lady Alice. As John and Alice Amaury have not lived at the castle for a long time, they don't speak Anglo-Norman to their sons William (who they named because they liked the name) and Amaury (who is named after John's father), and so the two boys don't learn the language. When John Amaury dies, his eldest son William inherits the family farm. As William can't speak Anglo-Norman, the name "Amaury" is tricky to pronounce, so he just becomes known as William Emery. His brother Amaury goes to work in town in Derby, but finds another Amaury from Leicester already living there. To distinguish himself from the other Amaury when the taxman comes round, Amaury decides to take a surname. He decides against Emery, because "Amaury Emery" is the silliest thing he's ever heard, but he thinks that "Amaury Johnson" after his father sounds good. Therefore, even though William Emery and Amaury Johnson are brothers, they have different surnames. William Emery's descendants will believe they are of Norman ancestry because of the Norman sounding "Emery", while Amaury's descendants will think they are Anglo-Saxon because of the Germanic suffix "-son". In fact, both William and Amaury are patrilineally descended from an Anglo-Saxon great-grandfather named Æthelwulf who was just trying to help his son climb the social ladder by giving him a Norman name.
Interestingly, a 2013 study showed that (in the UK) if you were in possession of a "Norman" surname (such as the heavy-hitters like Darcy, Percy, Mandeville, or Montgomery) you are more likely to be higher up the socio-economic scale with a greater chance of having attended Oxbridge in comparison to people with "Anglo-Saxon" surnames. These "Norman" surnames highlighted were often aristocratic ones connected to towns in Normandy (like Percy) and not ambiguous ones like Emery. It perhaps shows that there is something in a "Norman" surname indicating you are of Norman descent (as a father could pass his wealth and privilege onto his son), but at the same time, some names that appear Norman might not indicate your ancestry at all.
Short Bibliography