Hi Olandir,
Thank you for the question.
Here is the code to do that

add_action( 'bp_signup_validate', function () {

	$disallowde_names = array( "admin", "me" );

	if ( ! isset( $_POST['field_1'] ) ) {
		return;
	}

	if ( in_array( $_POST['field_1'], $disallowde_names ) ) {
		buddypress()->signup->errors['field_1'] = __( 'Sorry, this name is not allowed!' );
	}

} );

Please update the disallowed list with the names.

Please do understand that it does not prevent the users form changing the name after registration to one of the disallowed ones.

PS:- My colleague @ravisharma proposed another alternative code which works for update as well as registration(Here is his code)

add_filter( 'bp_xprofile_set_field_data_pre_save', function ( $save, $field_args ) {
    $field = $field_args['field'];
    $value = $field_args['value'];

    // Not name field.
    if ( $field->id != 1 ) {
        return $save;
    }

    $restricted_names = array( 'admin', 'moderator' );

    if ( in_array( $value, $restricted_names ) ) {
        $error_message = __( 'Name is restricted. Please try different name' );

        add_action(
            'bp_field_' . $field->id . '_errors',
            function () use ( $error_message ) {
                echo esc_html( $error_message );
            }
        );

        return new WP_Error( $error_message );
    }

    return $save;
},10, 3 );

The problem is it does not show the error to the user.

Regards
Brajesh

You are welcome.
You can use both of them if you want to.

In case you used the 2nd code, the message on update profile page will be “There was a problem updating some of your profile information. Please try again.” That’s a bit misleading.

Regards
Brajesh