Given you are only expanding the bore by 1/4" or so overall, and then replacing the lost material with a press fit steel bushing, it shouldn't appreciably affect the strength of the eye at all, unless it was very small, in which case you just make a new eye. If it's a simple cylindrical shaped eye, just make a new one and weld it on. If it is a forged eye on a big excavator boom cylinder or such, then you bush it as above.
Best technique on the boring and bushing is to bore first, then make your bushing. If your hole comes out a few thousandths too big or small, you can make the bushing to accommodate a lot easier than you can bore out another couple of thou in one pass.
My technique for boring fits is to just whack away until the last .050". Then one pass to take .020" and three at .010. The first .010 lets you work out the chatter and takes the spring out. The 2nd .010 pass is going to be honest, with no remaining spring to account for. Last .010 is to get it right on. You can make that pass anywhere from .007 to about .013 without having to worry about it missing by more than about .001.
The bigger the bore in the eye, the better a shrink fit works. If you have a 3" pin (say 75mm), you can take that out to 3 1/4" for the bushing (83-85mm) and be able to get about .015 expansion on the hole with 400 degrees F on the heat. Bore the eye 3.250 or there abouts. Make your bushing OD 3.255". You can easily shrink that in, then weld the sides with TIG to seal it in.
Now, if you leave the bore .050" or so under, you can line back up and bore the new bushing to the 3" OD for your original size bronze bushing. If it lands .002 too small on the last check, leave it alone and polish .002 off the OD of the bronze bushing. A lot easier than trying to skim .002 off the bore and risking it going a couple of thousandths over, leaving the bronze bushing loose.
