My question is straightforward:
What is the etymology of eh?
Could eh in Ich habe eh nichts zu machen be an abbreviation for eher?
(I spelled eh with an h because the e seems to be pronounced longer, more like [eˑ])
The words eh and eher share their origin
So, the word ēr, which does not exist anymore, means früh (early). This sense is still enclosed in the primary meaning of eher, namely at an earlier date. The other meanings of eher can chiefly be translated to rather or quite.
Regarding your second question the ultimate answer is 'No'. Eh is no abbreviation for eher and, furthermore, fulfills a completely different function. Its usage is colloquial, in the sense of anyway. For that reason, you can translate that sentence like this:
Without any context, however, it's hard to say if you should use eh or eher. I could think of a couple of situations where this sentence is fine with eher although the sentence will likely be slightly modified.
No. I would like to think that the "eh" here means "ohnehin". That is:
"Ich habe eh nichts zu tun."