It may be a corruption of a name in another language of ancient to medieval Middle Asia. In New Persian, ashkboos can be literally seen as "who kisses (boos) teardrop or Arsaces (both are ashk)". As far as I know, it first appeared in Shahnameh epics (~1000AD) as a Turanian hero who battled and lost to the Iranian protagonist, Rostam.
An Afghan friend of mine is named Ashkaboos. He did mention ancient Persia and something about "tears," if I remember correctly, "tears falling down." Perhaps this falling down of the tears was to the lips, and thus the kissing of the tears?
I think I need to make it clear again that the name may not be Persian, or any other languages close to Persian. In other words, the only certain Persian thing about this name is the language of the poet who composed the epic. Otherwise, almost certainly neither the language of the people who first told the story nor the language of Ashkboos himself (if he existed ever) was Persian. A significant number of epic names are from non-Persian or Old Persian origin. Though they look like a Persian word, their meaning wouldn't have been found if the original version of the names wasn't in old inscriptions or scriptures of their original language. For Ashkboos, it is probable that we simply don't have the original form of the name (or even the language).
Even if we fancy about a possible New Persian origin there are numerous possibilities:
- ashk+būs (teardrop + kiss), you are free to interpret this however you like whether metaphoric like you Afghan friend or literally as I did.
- ashk+kabūs (teardrop + skewed)
- arshak+būs (Arsaces + kiss). Arshak (shortened to Ashk in Persian) or Arsaces was the royal title of Arsacid kings of ancient Persia. They were of Parthians who are suspected to play an important role in developing epics like that of Ashkboos.
- arshak+būs (small bear / hero + kiss). arsh+ak (same as the proper noun Arshak)
P.S. The original pronunciation seems to be ending with bōs not būs. Later in both Dari and Persian the distinction between ō not ū has vanished. I'm not even sure we can talk about "kiss" as the meaning of the second part.