Broadly speaking, William the Conqueror's mission during the immediate post-Conquest period was to suppress the Anglo-Saxon nobility and bring their lands under Norman control in order to consolidate power under his rule. He was pretty systematic in how he converted English lands and titles to his Norman supporters over the two decades after Hastings, but that's not to say that the preexisting English culture (which was largely a mixture of various Scandinavian influences at this point) was totally overwhelmed.

For instance, there's a couple of entries in the Domesday Book that indicate that a few of the former English thegns managed to retain their holdings under William's rule. One such instance is a certain individual named "Thorkil de Cliveland" who resided in Yorkshire and is typically described as the father of Uctred, Earl of York c. 1066 and who was either the co-founder of Whitby Abbey, or had his lands confiscated by William c. 1086 and donated to the Abbey (sources are somewhat confused on this account). [1] What makes him interesting is the combination of the Scandinavian "Thorkil" with the Norman "de" prepended to the place name "Cliveland" because it indicated that the Scandinavian culture that flourished in York pre-Conquest was being subtly modified to suit Norman conventions without totally abandoning its roots.[2] The name Thorkil was passed down a number of times in successive generations of de Clivelands/Clevelands, with the last so-named recorded as residing in Ipswich in 1233. An individual by the name of "Thorkill of Arden" is also recorded in the Domesday Book as possessing land in Warwickshire, and there's a smattering of "Ulfs" as well.

I'd argue that the retention of Scandinavian names shows that there was at least some effort to "preserve" the pre-Norman culture. However, whether it was a subversive preservation of cultural identity, or if it was just that "Grandpa was named 'Thorkil' so we're naming our kid after him out of respect"... I don't know enough to make an educated guess one way or the other.

1. Puryear, Cynthia. The Effects of the Norman Conquest on Anglo-Saxon Aristocracy, pg. 15

2. It's entirely possible that Thorkil took the name "de Cliveland" immediately after the invasion in order to pledge himself to his new Norman overlords. He's the first de Cliveland recorded in England, so this is definitely something worth considering. Puryear seems to hint that this was the case.

You might also want to check out The Norman Conquest and the Genesis of English Feudalism by C.W. Hollister for more background on how the Conquest altered native English culture.