You can manage it by storing a selected button value in state and checking on disabled time. You can use the button disabled attribute for disabling. You can store unique value for matching button disable like name or id.

If we use name for selecting the button so we need to check like this way.

disabled={currentSelect && currentSelect !== element.name}

Example:

const projectTypeValues = [
  { name: "Hour", value: "hour" },
  { name: "Day", value: "day" },
  { name: "Session", value: "session" },
  { name: "project", value: "project" }
];
const App = () => {
  const [currentSelect, setSelect] = React.useState(null);
  console.log(currentSelect);
  return (
    <div>
      {projectTypeValues.map((element) => (
        <div style={{ padding: "10px" }} key={element.name}>
          <button
            disabled={currentSelect && currentSelect !== element.name}
            variant="contained"
            color="primary"
            onClick={() => setSelect(element.name)}
          >
            {element.name}
          </button>
        </div>
      ))}
    </div>
  );
};

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

<script
  crossorigin
  src="https://unpkg.com/react@17/umd/react.production.min.js"
></script>
<script
  crossorigin
  src="https://unpkg.com/react-dom@17/umd/react-dom.production.min.js"
></script>
<div id="root"></div>

import React from "react";
function App() {
    const [disable, setDisable] = React.useState(false);

 return (
        <button variant="contained" color="primary" 
         disabled={disable} onClick={() => 
         setDisable(true)}>
         {element.name} 
        </button>
         );
 }

You should define a state for the selected button, something like this:

const [selectedButtonIndex, setSelectedButtonIndex] = useState(-1)

then you need to use map and get the index and disable or other buttons:

 {projectTypeValues.map((element, index)=>
          <Button variant="contained" disabled={index !== -1 && index !== selectedButtonIndex} onClick={() => setSelectedButtonIndex(index)} color="primary">{element.name</Button>
 )}

Note: I started with -1 as initial value, as this enables all buttons initially and then it locks your choice. You could use another state to signal the fact that a choice has been made.