Just wondering if he got permission from the Serbian American's family.
He probably did but just curious.
Imagine of Elon had to change the name of his car company?
Don't want to even think of it.
He surely did not! The company existed and was named prior to his involvement. It was formed in July 2003 and Musk joined in February 2004.
Were the situation have been different it is not really clear that the name "Tesla" could be protected anyway, despite the obvious lineage. For a related example the 'tesla'=10K Gauss could be claimed to have had commercial intent also. It would be an entirely different question were the name to have been Edison, Firestone, Crapper or Ford all of which were explited commercially during the lifetime of the persons in question. In any event there would be a difficult case to make for the Tesla name since Nicola never married and had no progeny (OK, 'known progency', since there conceivably could have been).
Ergo...there is nobody from whom to seek permission, his name was not commercialised during his lifetime, so there is no potentially injured party to make such a claim. Thus, there was nobody from whom to seek permission.
This was an interesting question...