Not really a tenant, but help anyway

My brother's family had no income or place to live. My mother allowed them
to move into our home and paid for their furniture to be put in storage. It
is under her name. We had an extreme falling out, they abandoned their
property and left. How long do we have to store their possessions before we
can either sell it to recover some of the expenses they cost us, or throw
them away?

[OHlandlord]

This is hard to answer without knowing your state. Many states have state statutes on abandoned property. These statutes tell you how you need to notify the previous "tenant" (residents) that the property must be removed prior to disposal. Those statutes say the manner in which you must notify the tenant (in some places you can post to the door, in some you must mail to last known address, some you must publish a notice in the paper for a period of time, etc.).

They also say how long you must hold the possessions before you get rid of them, how you must dispose of them (some states say you must publically auction them), and what you must do with any proceeds from the sales of their possessions. Please post your state and I can give you the state statute number to read for the info you need.