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 which was built in 1947. We have remodeled almost all of it and added 1000 square foot in 2007. Lars Construction got me up to drywall and I did the rest.