Here are a few recommendations that have worked well in environments I've worked in the past :
Big +1 on what @val_baca said below. Determine alternate weeks and definitely have an on-call calendar that everyone's dates and rotations are on.
Documentation of critical systems and troubleshooting steps to get things back to good working state are key. This has saved hours of troubleshooting and downtime across multiple teams I've been in. Equally beneficial is person who's out can relax a lot more knowing things are being handled in a somewhat organized fashion in case of emergency, and they won't come back to a worse fire than the original one.
List out the major holidays/travel times of the year that people usually want to be with friends and family, e.g. Christmas, New Year's, etc. Figure out who's going to cover each one. It's often the worst feeling when everyone on a team has unchangeable plans over a long weekend and someone gets stuck with being on-call at the last minute.
Track on-call coverage data and adjust accordingly. Equitable scheduling should ideally balance out all incidents, but if a team deploys heavily at the beginning of every month, and the same person is always on-call during that time, chances are they'll get a lot more calls during off-hours than everyone else in the rotation. Assess who's getting "on-call" burn-out, and swap around the responsibilities for a few weeks to help them recover.
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Here are a few recommendations that have worked well in environments I've worked in the past :