What is the last possible time to exercise an equity option traded on CBOE?

It doesn't matter what the last time to exercise is because the OCC exercises all options that are one cent or more ITM at expiration (Exercise By Exception) unless the option owner provides a Do Not Exercise order to his broker. And since you are short the call, that has no relevance to you. You will be assigned if ITM.

With AAPL at $116, your $117.50 call is now 13 cents. You can watch it all day or you can roll it now. Option trading ends at 4 PM EST so that is the cut off time for rolling your option.

I would like to open another short call today. When could I roll my AAPL call without fearing that AAPL will surge beyond $117.50 today and my second short AAPL call induces a margin call due to being uncovered? 16:00 EST (end of trading)? 15:55 EST? I don't want to close the current short call. I want to let it expire worthless and avoid the commission.

There are several problems with this. First, rolling means closing the existing position and opening a subsequent position, either by legging out and in or by using a spread order to achieve both transactions in one fell swoop. The spread is preferable because you can work the price and you will not subject yourself to directional risk as you leg in/out.

Selling a second short call while the first one is open means that you have one naked call in order to earn 13 cents and avoid a commission is not a clever thing to do because options can be exercised until 5:30 PM. If AAPL surges, you'll pay a hefty price for that endeavor (loss on the naked call).

If you sell a naked call, you must meet the option's margin requirement (approximately 20%). Doing so will not create a margin call unless the unfathomable happens and AAPL skyrockets and you have insufficient margin in your account.

If you're going to chase 13 cents, I'd suggest that you either let the today's expiring call expire and sell the next call Monday morning or preferably, watch the position and let the 13 cents evaporate, assuming AAPL doesn't exceed $117.50 by 4 PM. Then, late in the day, use a spread order to roll to your next expiry.