PINE64

Would it be possible to somehow connect and run a GPU into a Pine 64 for cryptocurrency mining purposes?  

Any hardware that might be able to make a PCI-e 1x connection interface with this hardware?

No. But you could try something like this i suppose ?

edit: full disclosure, I know nothing about mining concurrency and have no clue is the guide I posted will work

eGPU-Docks ^^ now you need a thunderbolt connection at the PINE :p