How to change the styles of ListItem element with the "onclick" event?

Whenever you are trying to override Material-UI styles and it isn't working like you would expect, the best resource is the source code. Here is the URL for the ListItem source code: https://github.com/mui-org/material-ui/blob/master/packages/material-ui/src/ListItem/ListItem.js. For the most part, you only need to look at the styles variable near the top of the source file.

Below I've copied all of the portions of the styles variable that deal with backgroundColor and textDecoration:

export const styles = theme => ({
  /* Styles applied to the (normally root) `component` element. May be wrapped by a `container`. */
  root: {
    textDecoration: 'none',
    '&$selected, &$selected:hover, &$selected:focus': {
      backgroundColor: theme.palette.action.selected,
    },
  },
  /* Styles applied to the inner `component` element if `button={true}`. */
  button: {
    transition: theme.transitions.create('background-color', {
      duration: theme.transitions.duration.shortest,
    }),
    '&:hover': {
      textDecoration: 'none',
      backgroundColor: theme.palette.action.hover,
      // Reset on touch devices, it doesn't add specificity
      '@media (hover: none)': {
        backgroundColor: 'transparent',
      },
    },
    '&:focus': {
      backgroundColor: theme.palette.action.hover,
    },
  },
  /* Styles applied to the root element if `selected={true}`. */
  selected: {},
});

The main styles causing difficulty are the button hover and focus styles. In order to override these successfully without resorting to "!important", you need to have the appropriate CSS specificity.

The following seems to do the trick:

  listItemDone: {
    borderRadius: "1em",
    "&,&:focus,&:hover": {
      backgroundColor: "#F6F6E8",
      textDecoration: "line-through"
    }
  },

However, the above prevents there from being any hover effect on "done" items, so you may instead want to do something more like:

  listItemDone: {
    borderRadius: "1em",
    "&,&:focus": {
      backgroundColor: "#F6F6E8",
      textDecoration: "line-through"
    },
    "&:hover": {
      textDecoration: "line-through"
    }
  },

This allows the hover background color of done items to still be theme.palette.action.hover. If you want the hover color to be different for done items, you can specify it explicitly along with the textDecoration.

There is one other detail to take care of. If you click a list item to put it in a "done" state and then click it again, it will not be in the "done" state anymore but it will have the button focus styles applied. In order to remove that focus styling, you also need the following:

  listItem: {
    borderRadius: "1em",
    "&,&:focus": {
      backgroundColor: theme.palette.background.paper // or whatever color you want this to be
    }
  },

Here is my modified version of your sandbox:

So I've tried to add !important to this styles and it started working as expected:

  listItemDone: {
    borderRadius: "1em",
    backgroundColor: "#F6F6E8 !important",
    textDecoration: "line-through !important"
  },

demo. It looks like this material-ui just overrides your styles (textDecoration and backgroundColor) when you hover over elements. Hope that helps.