First of all, the span element with the click event needs to have a name property otherwise, there will be no name to find within the e.target. With that said, e.target.name is reserved for form elements (input, select, etc). So to actually tap into the name property you'll have to use e.target.getAttribute("name")

Additionally, because you have an array of objects, it would not be effective to use list.indexOf(e.target.name) since that is looking for a string when you are iterating over objects. That's like saying find "dog" within [{}, {}, {}]

Lastly, array.slice() returns a new array starting with the item at the index you passed to it. So if you clicked the last-item, you would only be getting back the last item.

Try something like this instead using .filter(): codesandbox

import React, { useState } from "react";
import ReactDOM from "react-dom";

import "./styles.css";

const App = () => {
  const defaultList = [
    { name: "ItemOne" },
    { name: "ItemTwo" },
    { name: "ItemThree" }
  ];

  const [list, updateList] = useState(defaultList);

  const handleRemoveItem = (e) => {
   const name = e.target.getAttribute("name")
    updateList(list.filter(item => item.name !== name));
  };

  return (
    <div>
      {list.map(item => {
        return (
          <>
            <span name={item.name} onClick={handleRemoveItem}>
              x
            </span>
            <span>{item.name}</span>
          </>
        );
      })}
    </div>
  );
};

const rootElement = document.getElementById("root");
ReactDOM.render(<App />, rootElement);

You can use Array.filter to do this in a one-liner:

const handleRemoveItem = name => {
    updateList(list.filter(item => item.name !== name))
}

Eta: you'll also need to pass the name of your item in your onClick handler:

{list.map(item => {
    return ( 
        <>
        <span onClick={() =>handleRemoveItem(item.name)}>x </span>
        <span>{item.name}</span>
        </>
    )}

I faced the same issue and came up with a solution using the answers to this question. I'm putting my final solution here because I had to use multiple answers to get there.

const defaultList = [
    { name: "ItemOne" },
    { name: "ItemTwo" },
    { name: "ItemThree" }
]

const [list, updateList] = useState(defaultList);

const handleRemoveItem = idx => {
    // assigning the list to temp variable
    const temp = [...list];

    // removing the element using splice
    temp.splice(idx, 1);

    // updating the list
    updateList(temp);
}

return (
    {list.map((item, idx) => (
      <div key={idx}>
        <button onClick={() => handleRemoveItem(idx)}>x </button>
        <span>{item.name}</span>
      </div>
    ))}

)