add_action('frm_after_create_entry', 'after_saved_check_other', 30, 2);
add_action('frm_after_update_entry', 'after_saved_check_other', 10, 2);
function after_saved_check_other($entry_id, $form_id) {
	if ($form_id == 4) { // change 4 to your form's id
	    $fieldId = 10; // change 10 to your checkbox's field id
		$values = $_POST['item_meta'][$fieldId]; 
		// NOTE: for multi-select fields the $val is an array that can look like this Array ( [0] => Option 1 [other_2] => Other )
		//  so, the values for Dropdowns, Radio Buttons and Checkboxes should be handled differently 
		//  since in the case of a checkbox, "Other" could be one of the choices, while in the case of a 
		//  radio button there's only one choice
		// For this example, I will handle the more complex case of a checkbox
		foreach ($values as $key => $option) {
			if ($option == "Other") {
				$otherKey = $key;
				$newVal =  array_values(FrmEntryMeta::get_entry_meta_by_field($entry_id, $fieldId))[0];
				$newVal = sanitize_text_field($newVal);
				global $wpdb;
				$sql = "SELECT options FROM {$wpdb->prefix}frm_fields WHERE id=%d";
				$sql = $wpdb->prepare($sql, $fieldId);
				$options = $wpdb->get_var($sql);
				$options = unserialize($options); // convert to a format that's easier to manipulate
				
				// if the Other value is not already in the array, add it
				// NOTE: You may choose to do something to the new value such as camel-case it here
				if (!in_array($newVal, $options)) {
					unset($options[$otherKey]);
					// piece together the new array (btw, I tried array_slice and couldn't get it to work)
					$newOptionArr = array("label" => $newVal, "value" => $newVal);
					array_push($options, $newOptionArr);
					$options[$otherKey] = "Other";
					$options = serialize($options); // convert back to format for saving
					
					$sql = "UPDATE {$wpdb->prefix}frm_fields SET options='%s' WHERE id=%d";
					$sql = $wpdb->prepare($sql, $options, $fieldId);
					$wpdb->query($sql);
					// the following is necessary to clear the cache for the change to take place immediately
					FrmField::delete_form_transient( $form_id );
				}
				break; // drop out because there can be only one Other choice
			}
		}
	}
}