How to verify if value already added to the list and if so, then update the value?
user = User.get_by_key_name(user_key_name)
if user:


Python convention is to use user is None when checking against None

    field_names = user.field_names
    field_values = user.field_values
    field_scores = user.field_scores


Why copy these references? It doesn't help you much.

    if field_name in user.field_names:
        # field_names[field_index] = field_name
        field_index = user.field_names.index(field_name)


Use index to check if field_name is present. That way it doesn't have to search through the lists twice.

        field_values[field_index] = field_value
        field_scores[field_index] = field_score


If possible, avoid maintaing parrallel lists. In this case, it looks like maybe you should have a dictionary of strings to tuples.

    else:
        field_names.append(field_name)
        field_values.append(field_value)
        field_scores.append(field_score)
    user.field_names = field_names
    user.field_values = field_values 
    user.field_scores = field_scores


Unless user does something odd, this has no effect. Assignment doesn't copy in python, it just copies the references. 

    user.put()