"someone nodded abstractly"
One of the meanings of "abstract" given in the OED (though marked as "archaic") is:
- Withdrawn from the contemplation of present objects; = abstracted adj. 2. arch.
That is how I would understand "nodded abstractly" - "nodded as though with his mind on other things".
As Colin quoted, the use of abstractly in the sense "absent-mindedly" is now archaic—I don't remember ever seeing it. Such an archaic word would not fit the rest of the text. It is most likely that this writer intended abstractedly, which means the same and is in current use.
Distractedly might be what they mean.
The adjective abstract is defined here as:
5. Impersonal, as in attitude or views
An Ngram shows that nod abstractly and nodding abstractly seem to never be used, and nodded abstractly only began to be used recently (a little in the 1930's and increasingly in the 1980's and 1990's). I agree that it probably derives from mistaking the word with distractedly.
Given all of that, I expect it would refer to an impersonal nod, a simple acknowledgement of another.
I did a quick Google and most of the instances of this I found were in fan fictions and amateur short stories. I'm not sure it's a proper turn of phrase, probably someone misheard Nodded Distractedly and got it wrong.
It just doesn't make sense...