I forgot that Picard also has an archaeology hobby as well-that might be something they could look into. The second season trailer of Discovery, interestingly enough, seems to have a bit of "The Chase" element to it (Hidden messages spread across the galaxy etc).
Also it'd be nice to see Beverly back. Her relationship with Picard was kind of growing in the last season or so, but the movies pretty much abandoned it. (The novels did have them eventually marry and have kids though).