Can't you just do score={50} once?

const Header = (props) => {
  return (
    <header>
      <h1>{props.title}</h1>
      <span className="stats">Players: { props.totalPlayers }</span>
    </header>
  );
}

const Player = (props) => {
  return (
    <div className="player">
      <span className="player-name">
        { props.name }
      </span>

      <Counter score={35}/>
    </div>
  );
}

const Counter = (props) => {
  return (
    <div className="counter">
      <button className="counter-action decrement"> - </button>
      <span className="counter-score">{ props.score }</span>
      <button className="counter-action increment"> + </button>
    </div>
  );
}

const App = () => {
  return (
    <div className="scoreboard">
      <Header title="scoreboard" totalPlayers={1} />

      {/* Players list */}
      <Player name="Guil" />
    </div>
  );
}

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