You have a few syntax errors in your code. Try this:
$stmt = $dbConnection->prepare('INSERT INTO users (name) VALUES (:s)');
$stmt->bindParam(':s', $name);
$stmt->execute();
If you want to insert and define more values, do it like this:
$stmt = $dbConnection->prepare('INSERT INTO users (name, email) VALUES (:s, :email)');
$stmt->bindParam(':s', $name);
$stmt->bindParam(':email', $email);
$stmt->execute();
If you're using mysqli, your code will look like this:
$stmt = $dbConnection->prepare('INSERT INTO users (name) VALUES (?)');
$stmt->bind_param('s', $name);
$stmt->execute();