I think it would be cool in the second film if there was a scenario where McCoy's divorce could somehow serve to upend all that he's worked for to this point. (vindictive b*tch of a wife? child with daddy issues? some heretofore undiscovered issue in the divorce case?) It doesn't have to be a major plot point. Indeed, it could be a sub-plot. If woven in just right, it can help with the main tale, and perhaps serve to make it even more compelling.
Or perhaps Spock dealing with more of the racial issues among his people, despite the cataclysm that befell them in the first film. Something where Spock points out that most Vulcans have learned nothing since the disaster.
Or a sub-plot (or shorter teaser story) where Kirk returns to Earth, only to find a not-so-impressed stepfather who still seems, somehow, to try and run Kirk's life into misery...since stepdaddy seems to be miserable himself. Perhaps Kirk's mother dies, and when he and his brother return home to take care of the funeral arrangements, he comes into conflict with a stepfather who somehow blames the siblings for his mother's death. (Broken heart leading to failing health, and a loss of the will to live, or so the stepfather indignantly and pettily claims...when in truth, she could not have been prouder of her children--perhaps Kirk's older brother took on a profession that also made his mother proud, but stepdaddy ain't gonna tell him that.) There could even be some kind of slight sibling tension....but nothing near the hostility that the Kirks' stepfather holds toward them both.