Nobody in the family thinks about the water softener until the shower starts feeling like a bar of soap that will not rinse. By then the problem is rarely the hardware. It is that nobody was assigned to the salt.
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Line item one: the salt
Most softeners of the common kind run on salt pellets or blocks stored in a brine tank. Check the owner's manual or the label on the unit for the type it takes. Then decide who buys it, where it is stored, and who carries it down to the utility room. Bags are heavy. If you are the one carrying them, that is labor, and the family should count it as labor, not a favor.
Line item two: the tank level
Salt runs down out of sight. Many units have a window or a float that shows the level. Ask who will look at it and how often. A simple rule works: the person who looks at the level also writes the date of the last fill on a card taped inside the cabinet door. Keep the card on paper where any sibling can read it, not only on your phone.
Line item three: the settings and the service
Your mother probably did not set the hardness dial, and you may not know what it reads. Find the setting in the manual, photograph it, and put the photo in the shared folder next to the card. If a service visit is ever needed, the person who calls the technician should be the person named on the card, so the record and the call live in one place.
Who owns what
Write three names next to three jobs: buys salt, checks the level, keeps the record. Split them if you like, but make sure each job has exactly one name. Shared jobs with no name are how a softener quietly stops working, and how the first person who notices ends up holding the whole thing.
Keep the list short enough to read from the utility room door. A fair arrangement is one that everyone can see from the stairs.