Currently, we're just using Google Keep, since it at least supports syncing, sharing, and checklists, but I'm hoping to find something more privacy friendly, and/or more feature rich. In particular, what I'm looking for, in order of priority, are as follows:
Doesn't need to be self-hosted
Lists that can be shared and synced across users and devices
Basic Checklist functionality
Reminders, ideally with options to repeat until complete or similar
Automation, such as automatically recurring events, esspecially recurring after date of completion
Private storage of user data, or at least better than Google
I'd recommend either Peppermint OS or Lubuntu as lightweight distros for an old laptop. There are many more distros that you can try out there, even without desktop environments (no graphical stuff, which saves a lot of resources), if you're not into using your laptop for stuff outside of server-use. For that, Debian is the go-to imho. Just remember not to select any desktop environment during the installation process.
Put some of the distros on a flash drive and try them out. Distro hopping can be fun :-)