You'll find an other version of this hadith in sunan abi Dawod, compiled in the book of general behavior in the chapter entitled changing bad names. And also in Sahih al-Bukahri in the book of good manners in a chapter entitled to change a name for a better.
The first and most important thing we can learn is that it is allowed and recommended to change bad names, but it isn't obligatory. You may find this in many hadith commentaries:
Ibn Battaal said (in his commentary on Sahih al-Bukhari):
This shows that improving names and changing names to something better is not something that is obligatory.
Ibn Hajar wrote in Fath al-Baari:
And if it had been obligatory, the Sahaabi would not have refused to change his name, and the Prophet (peace and blessings of Allaah be upon him) would have compelled him to change his name.
It is necessary to say that the fact that this roughness (الحزن) in their behavior was a characteristic in the family of Sa'id ibn al-Musayib shouldn't be overrated as an influence of the naming on the character, yes there might be a relationship, but how can this go ahead with words of the Quran such as (13:11):
... Indeed, Allah will not change the condition of a people until they change what is in themselves. ...
which imply that we are able to change things and conditions. And with many ahadith teaching us how to become better people!
The fact that the Prophet () in this case didn't reject or oppose the answer can be explained two ways: Either it is generally not obligatory, but in some versions of the hadith (see the link from sunan abi Dawod) the grandfather replayed:
No, smooth is trodden upon and disgraced
which was also qualified as a kind of arrogance so his offspring having a kind of roughness in their manners can also be considered as a punishment for his arrogance. As al-Qarri added as a side note in his Mirqat al-Mafatih.
A bit off-topic
The hadith was also compiled in other hadith collections such as Musnad Ahmad, Sahih ibn Hebban, as-Sunan al-Kubra and al-Adab of al-Bayhaqi (with the same wording of al-Bukahri) beside other collections such as at-Tabaqat of ibn Sa'ad and at-Tabarani's al-Mu'jam al-Kabir (with 3 different chains with some differences in the wording) etc..
Note that one could say that this hadith doesn't have a high sanad (short narrator chain) as both the father of Sai'd ibn al-Musayyib and his grandfather (died in the battle of Yamama) were sahaba and have converted to Islam in the year of fath Mekka (arabic wikipedia says they were among those who made the pledge of the tree)! However in some of the collections it was either directly narrated by Sai'd ibn al-Musayib from his father or from his grandfather. This seems rather the case for collections written by historians such as ibn Sa'ad or al-Asbahani and at-Tabarani as one of the rare muhhadetheen whom have narrated it with a shorter chain. Ibn Hajar however says that the long chain is more correct while anything else would be a mursal!
Finally there's a du'a which is qualified as sahih by al-Albani, it is quoted in Sahih ibn Hebban using both (hazn) and (sahl):
اللهم لا سهل إلا ما جعلته سهلا وأنت تجعل الحزن إذا شئت سهلاً
Allahumma laa sahla illa ma-ja'altahu sahlan wa anta taj'alu al-Haz(a)na idha shi'ta sahla(n).
“O Allaah, nothing is easy (sahl) but that which You make easy and You can make hardship (hazn) easy if You will.” (source)
Some English references islamqa #14622 and #14626