Find How Many Times Numbers Repeat on a Digital Clock
Input loop

Your

        print("Please enter an integer.")


can probably be lumped into the input statement after it.

Time math

Usually doing your own should be avoided. As an example, this:

total_minutes = (24 * 60) / 2


can be

from datetime import timedelta
# ...
total_minutes = timedelta(hours=12) / timedelta(minutes=1)


General approach

Currently you're doing a lot of string manipulation, including a stringly-typed hours and minutes in gen_times. Your code would be simpler if, instead of maintaining and incrementing those strings, increment a single datetime.time instance. From there, to get individual digits, you could  strftime it to a string and feed that to a Counter. The strftime format string in this case would be most useful as '%I%M'.

Variable names

Try to avoid names like i and ii. In particular, this:

    while(ii &lt; 10):
        count = times[i].count(str(ii))
        #iterationcounter+=1
        if count &gt;= int(input_repeats):
            counter_repeat+=1
            break

        ii+=1


is better represented as

for digit in range(10):
    count = times[i].count(str(digit))
    if count &gt;= int(input_repeats):
        counter_repeat+=1
        break


Though that times[i].count should be replaced with the use of a Counter.