Upvote:6
The practice seem more or less untestified outside the Eyrbyggja saga. The saga deals with religious matters to an unusual degree, and is in fact one of the more importantsources for religious practices, even if it was written by a Christian long after the events and should be treated carefully. Thus, questions about how common this was can not be answered.
As for special status, my impression is that Thorsteinn was expected to be friend of Thor in a similar manner as Thorolf had been. Compare how "Thorolf" was actually named "Rolf" (chapter 3). Also, see how he named the isthmus "Thorsnaess" without it taking on any deeper significance, but the mountain he thought was sacred and surrounded with taboos, he called "Helgafjell", "holy mountain" (chapter 4).
The names starting with Thor were, however, unusually common in that family. I believe this can be compared with similar modern practices: while you would not necessarily think someone named "Grace" had particularly devout parents, if all siblings had virtue names, the odds are different.
Apart from the Eyrbyggja saga, in Swedish translation by Mikael Males, I've consulted Britt-Mari NΓ€sstrΓΆm Fornnordisk religion and H R Ellis Davidson's Gos and Myths of Northern Europe; while both discuss some aspects of the saga, the dedication of Thorsteinn is not one of them.