How can I repair a tiny puncture in leather?
Use a toothpick to push a small square or circle of backer fabric through the hole to the back of the leather. Choose something thin but strong and that has a little give - silk is often used but not everyone has a tiny bit of silk in their sewing basket. This gives some support for step 2. 
Use same toothpick to pick up a small dab of flexible, clear fabric glue through the hole to the backer fabric behind. 
The use a spoon to push the leather onto the fabric backer and rub it down to ensure the glue is well and good squished behind and around the hole, and that there are no edges to the tear that might catch again. If you can push the leather together you can make the hole practically invisible. No glue should be visible from the outside except the tiniest amount inside the seam itself.