A reissue application requires an application data sheet (ADS). I have never tried, but I suspect that when the USPTO issues the reissue patent, they will list inventors in the order of names provided in the ADS filed with the reissue application.
See USPTO MPEP Section 1410.
If not, you should be able to change the order of inventor names with a certificate of correction (and appropriate fee).
See USPTO MPEP 1481.02.
The USPTO will not issue a new ribbon copy for a certificate of correction, so the inventors will not see the first name change on the face of the patent. I suspect that Google Patents will not pick up the change, either. So, the result of a correction might be unsatisfactory for the inventors.
A certificate of correction is also the way to go if an inventor changes name, such as through marriage.