Of course there are Asturian names, but most people living in Asturias give their children regular Spanish names. A lot of Asturian names sound very similar to normal "Castillian" names, since they come from the same root. For example, the Spanish
Juan has the Asturian equivalent of
Xuan, much like there is the equivalent
Joan in Catalonia. There has been a revival of Asturian names since the nineties, so people with names like Xelu or Xuacu tend to be either very old or little kids, although there are some exceptions, like Olaya and Xana, that have been popular for longer. My father, for example, is called Nel (Asturian for
Manuel). His given name at birth was
Manuel, because non-Spanish names weren't allowed under
Franco's dictatorship, but he had it changed when it became legal.
If you search for "nombres Asturianos" or "nomes Asturianos" online you should find quite a lot of lists.