I think maybe some of the medicine used for lyme (both chronic or acute) can trigger MG. It can cause MG to reveal itself.

And another thing; chronic infections may contribute to the existence of auto immune diseases like RA or MG. Not sure id this is proven yet but it just seems logical to me. Having your immune system work overtime for months, even years.

And of course, both have symptoms that arent always clear, and they both can take years to diagnose.

But I don't think there is much else they have to do with eachother, just like a bunch of other diseases that fit the discription above (but i'm no expert and can be wrong of course )