I just move to a house with a double garage (~18' wide) and to save some space I have started to build a lift for my small utility trailer (300 lbs). In my previous garage that was only 10' wide I used 4X4s and put eye bolt right through the 4x4 vertically and it worked great to lift my trailer out of the way. Knowing I was going to need something a lot beefier I have installed two pairs of 2X10 running widthwise and spaced 5 ' apart of course on a separate load bearing structure and not the garage walls :-).
Now I am not sure how to fasten my pulley system to the 2X10s which is now really just two beams 4X10. I had thought about using a cable around the beam or even putting the eye bolts through horizontally but that would put twist on the beam.
Based upon from what I understand from your post, I would run a 4x4 perpendicular to your joists and then bolt through that with a big fat washer on top. That way you have some meat to tie into instead of running a lag bolt into the joist.
My constant project is my home, my play time is usually spent on my CNC router.