What can be done about a weapon's charge to clear a criminal record? 

Unless the conviction just came through in the last month or so, it is far, far too late for an appeal.

It's possible, in theory, to get a conviction vacated after the fact, but usually that's only done in cases where new evidence, that wasn't available at trial, makes it very, very clear that the verdict was wrong.  The defendant (your husband) changing his story, even if there's a persuasive reason to believe that the new testimony is the truth, is almost never good enough, without something to back it up (such as your husband's brother coming clean as well).

In this case, it might not have made any difference anyway.  Since the car was your husband's, the law usually is that he's "in possession" of anything that's in the car, if it isn't being physically carried by a passenger (and sometimes, even if it is).  So, even if in "reality" he was innocent, legally he was guilty.