html5 oninvalid doesn't work after fixed the input field

If you set a value with setCustomValidity() then the field is invalid. That is setting a non-zero length string causes the browser to consider the field invalid. In order to take effects of your new input you have to clear the custom validity. Simply you can use the following:

<input required maxlength="255" id="information_name" minlength=1 name="information[name]" oninvalid="setCustomValidity('Should not be left empty.')" 
   oninput="setCustomValidity('')" />

There is no need to use Javascript for this.

You can set a custom validity message with setCustomValidity, however any non-blank string will cause the field to act as if it had an invalid value. You need to setCustomValidity('') to clear the invalidated state of the field.

If your validity is simple and controlled via field attributes, you can use object.willValidate to do the test and set the custom validity message:

oninvalid="this.setCustomValidity(this.willValidate?'':'your custom message')"

If you setCustomValidity to any value that is not the empty string then that will cause the input to be in the invalid state. So your condition is checking for a value, then setting the field to be invalid either way. Only setCustomValidity when the value in the field is actually invalid, otherwise set it to an empty string:

<script>
 function check(input) {
   if (input.value == "") {
     input.setCustomValidity('Debe completar este campo.');
   } else  {
     input.setCustomValidity('');
   }
 }
</script>