Don't use a drill bit, it will grab and twist and make a ragged hole. You need to get a hole punch. You can get sets of them for not a lot of money. Then you get a piece of scrap wood; lay the strap on the wood with the rough side down; and grab a decent hammer. Then place the leather punch on the top of the strap in the location you want to place a hole and smack it hard enough with the hammer to cut through the leather and slightly into the scrap wood.. Twist and pull the punch out using the end of the punch you smacked with a hammer to remove it leaving a very professional looking clean and precise hole in your strap.
Here's a link to one inexpensive set.
This is a leather punch set available at Amazon for $7.99. It includes six sizes (1/8", 5/32", 3/16", 7/32", 1/4", 5/16"). Measure the holes that are already in your strap to find which one to use.
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The slot on the side is so you can remove the leather pills removed from the leather to make the hole. If making multiple holes it will continue to slide them out and out. I have a set like this I've used for leather work and it's simple and looks good. You could also buy a leather hole punch pliers but they are a LOT more expensive.