Here's a complete example that loads the JSON file, removes the target object, and then outputs the updated JSON object to file.

#!/usr/bin/python                                                               

# Load the JSON module and use it to load your JSON file.                       
# I'm assuming that the JSON file contains a list of objects.                   
import json
obj  = json.load(open("file.json"))

# Iterate through the objects in the JSON and pop (remove)                      
# the obj once we find it.                                                      
for i in xrange(len(obj)):
    if obj[i]["ename"] == "mark":
        obj.pop(i)
        break

# Output the updated file with pretty JSON                                      
open("updated-file.json", "w").write(
    json.dumps(obj, sort_keys=True, indent=4, separators=(',', ': '))
)

The main point is that we find the object by iterating through the objects in the loaded list, and then pop the object off the list once we find it. If you need to remove more than one object in the list, then you should store the indices of the objects you want to remove, and then remove them all at once after you've reached the end of the for loop (you don't want to modify the list while you iterate through it).

First question

However, whenever there's more than two elements and I enter anything higher than two, it doesn't delete anything. Even worse, when I enter the number one, it deletes everything but the zero index(whenever the array has more than two elements in it).

Inside delete_data() you have two lines reading i = + 1, which just assignes +1 (i.e., 1) to i. Thus, you're never increasing your index. You probably meant to write either i = i+1 or i += 1.

def delete_data():    # Deletes an element from the array
    view_data()
    new_data = []
    with open(filename, "r") as f:
        data = json.load(f)
        data_length = len(data) - 1
    print("Which index number would you like to delete?")
    delete_option = input(f"Select a number 0-{data_length}: ")
    i = 0
    for entry in data:
        if i == int(delete_option):
            i += 1  # <-- here
        else:
            new_data.append(entry)
            i += 1  # <-- and here

    with open(filename, "w") as f:
        json.dump(new_data, f, indent=4)

Second question: further improvements

Is there a better way to implement that in my Python script?

First, you can get rid of manually increasing i by using the builtin enumerate generator. Second, you could make your functions reusable by giving them parameters - where does the filename in your code example come from?

# view_data() should probably receive `filename` as a parameter
def view_data(filename: str):   # Prints JSON Array to screen
    with open(filename, "r") as f:
        data = json.load(f)
        # iterate over i and data simultaneously
        # alternatively, you could just remove i
        for i, item in enumerate(data):
            name = item["name"]
            chromebook = item["chromebook"]
            check_out = item["time&date"]
            print(f"Index Number: {i}")
            print(f"Name : {name}")
            print(f"Chromebook : {chromebook}")
            print(f"Time Of Checkout: {check_out} ")
            print("\n\n")
            # not needed anymore: i = i + 1

# view_data() should probably receive `filename` as a parameter
def delete_data(filename: str):    # Deletes an element from the array
    view_data()
    new_data = []
    with open(filename, "r") as f:
        data = json.load(f)
        data_length = len(data) - 1
    print("Which index number would you like to delete?")
    delete_option = input(f"Select a number 0-{data_length}: ")
    # iterate over i and data simultaneously
    for i, entry in enumerate(data):
        if i != int(delete_option):
            new_data.append(entry)

    with open(filename, "w") as f:
        json.dump(new_data, f, indent=4)

Furthermore, you could replace that for-loop by a list comprehension, which some may deem more "pythonic":

new_data = [entry for i, entry in enumerate(data) if i != int(delete_option)]

Let's assume you want to overwrite the same file:

import json

with open('data.json', 'r') as data_file:
    data = json.load(data_file)

for element in data:
    element.pop('hours', None)

with open('data.json', 'w') as data_file:
    data = json.dump(data, data_file)

dict.pop(<key>, not_found=None) is probably what you where looking for, if I understood your requirements. Because it will remove the hours key if present and will not fail if not present.

However I am not sure I understand why it makes a difference to you whether the hours key contains some days or not, because you just want to get rid of the whole key / value pair, right?

Now, if you really want to use del instead of pop, here is how you could make your code work:

import json

with open('data.json') as data_file:
    data = json.load(data_file)

for element in data:
    if 'hours' in element:
        del element['hours']

with open('data.json', 'w') as data_file:
    data = json.dump(data, data_file)

EDIT So, as you can see, I added the code to write the data back to the file. If you want to write it to another file, just change the filename in the second open statement.

I had to change the indentation, as you might have noticed, so that the file has been closed during the data cleanup phase and can be overwritten at the end.

with is what is called a context manager, whatever it provides (here the data_file file descriptor) is available ONLY within that context. It means that as soon as the indentation of the with block ends, the file gets closed and the context ends, along with the file descriptor which becomes invalid / obsolete.

Without doing this, you wouldn't be able to open the file in write mode and get a new file descriptor to write into.

I hope it's clear enough...

SECOND EDIT

This time, it seems clear that you need to do this:

with open('dest_file.json', 'w') as dest_file:
    with open('source_file.json', 'r') as source_file:
        for line in source_file:
            element = json.loads(line.strip())
            if 'hours' in element:
                del element['hours']
            dest_file.write(json.dumps(element))

It is usually a bad idea to delete from a list that you're iterating. Reverse iterating avoids some of the pitfalls, but it is much more difficult to follow code that does that, so usually you're better off using a list comprehension or filter.

obj["_notes"] = [x for x in obj["_notes"] if x["_type"] != 1]

This gives us the expected output :

{'_notes': 
   [
       {
            '_time': 10, 
            '_lineIndex': 2, 
            '_lineLayer': 0, 
            '_type': 0, 
            '_cutDirection': 7
       }
   ]
}

Use a simple filter:

no_percent = lambda feature: '%' not in feature['properties']['name']
data['features'] = filter(no_percent, data['features'])

Or as a list comprehension:

data['features'] = [feature for feature in data['features']
                      if '%' not in feature['properties']['name']]

You will have to read the file, convert it to python native data type (e.g. dictionary), then delete the element and save the file. In your case something like this could work:

import json

filepath = 'data.json'
with open(filepath, 'r') as fp:
    data = json.load(fp)
del data['names'][1]

with open(filepath, 'w') as fp:
    json.dump(data, fp)

An easy solution to your problem is deleting the unwanted key in place, with del:

import json

with open('data.json') as json_data:
    data = json.load(json_data)
    for element in data: 
        del element['imageData'] 

You should add some safety checks, but you get the idea.

One of approaches from Remove element from list when using enumerate() in python - a loop over a copy of the list referred as [:] - works too:

for a in myData['myArray'][:]:
  if (a['name'] == u'bad'):
     myData['myArray'].remove(a)

Thanks everyone!