Syncellus theorizes that it means Edomite: ιδουμαιος
Georgius Syncellus, Ecloga chronographica, p. 413 line 27
A.A. Mosshammer, Georgius Syncellus. Ecloga chronographica, Leipzig: Teubner, 1984: 1-478.
Damneion though is a much different word than Idumean. I think that it is not the meaning. But it shows that Syncellus did not take the name to actually be a real normal first name of Damneion.
Also, thanks (to BPK) for confirming that it's a Hapax, never showing up elsehwere in Josephus.
Benjamin Kantor, BPK , said it looks like Subduer in the chat box.
(" It is difficult to know with so little information. It has a nominal morpheme ending on it, naturally, so perhaps something like 'subduer' yes ")
A analogous name that Ben gave is νικητης meaning victor (feminine)/
So based on this response, it looks like a real name or honorific that his father could have been given.