After the Völsungs my father wrote (the Chosen), but struck this out. An etymological speculation on the origin of the name which (at any rate at one time) he favoured associated it with the Germanic words meaning "choose". Struck it out, right. Because, far as I can see, Völsung comes from Völsi, which means 'prick'. Here's Andy Orchard on that latter word:
Völsi. A horse penis worshipped as part of a cult practice ... when a pagan farmer's horse dies they eat the flesh, which was held to be taboo to Christians, and preserve the penis. which becomes the object of veneration, after the farmer's son has first waved it in front of the ladies, saying:
Here you can see
a rather big plonker,
one sliced off
the horse's dad.
Slave-girl, for you,
this völsi will be
none too sluggish
between your thighs. 
Orchard's translation, presumably ('plonker'); and what a charming farmer's son thereby portrayed.
This seems a rather obvious root for the family name at the heart of the saga: a tale of phallic productivity and assertiveness, with a great deal of to-do concerning 'swords' and 'rings', all tending to demonstrate that although it enjoyed its moments of triumph, the seed of the Völsi is doomed eventually to die.