Re-cord when sewn into the side seam
It's not uncommon for drawstrings to come loose and disappear. If the casing in the waistband is open all the way around, you can use a safety pin or other tool to thread a new drawstring through, but what if it only goes halfway around?
You can also use this technique to insert a drawstring if there wasn't one before!
I'll write some suggestions about that at the bottom of the post!
That was the case with these shorts, but with a good seam ripper, a sewing machine, and some pins, it was very easy to fix, and it was faster than buying new ones!
Rip open the waistband all the way to where the drawstring is attached on both sides. Since it's elastic and gathered, it's best to use the sharp part of the seam ripper and proceed carefully. On straighter seams, it's easier to use the side with the red knob down into the seam and just cut along, but that wasn't as easy here.
This is what the shorts looked like with the front part of the waistband ripped open. The drawstring is sewn into the sides, so a bit of ripping is needed there too, enough to pull out the stuck drawstring (if there still is one) and to insert a new one.
I had a shoelace lying around from some time, but other round cords, ribbons, or anything you find can be used too! You could even sew a fun cord if you want, while you're fixing things up (in that case, I can recommend the Roleau turner for turning a narrow, wrong-side-out band, for example).
I chose to make the drawstring a bit longer than it was, as children's clothing often has very short drawstrings for safety reasons. (There are strict production rules for such things.)
Push the end of the drawstring well into where it needs to be sewn. The one that came loose probably wasn't far enough in.
Sew across the waistband, elastic, and drawstring. Sew over a couple of times to ensure it's securely attached.
Thread the drawstring through the buttonhole. If you sew yourself, it's worth noting that the buttonhole is reinforced with an extra layer of fabric; it holds its shape better and is easy without the need for additional materials like interlining.
Before you sew the waistband shut, it might be a good idea to put in some pins. Make sure the cord is far away from where you are going to sew. Pins help to keep the cord out of the way. 
This is what it looked like before I sewed it up. 2-3 pins on each side, and the drawstring sewn firmly into the side seams and threaded through the buttonholes. 
Sew from the wrong side if you can, but the right side is fine. Make sure to pull the elastic so that the fabric you are sewing on is straight and smooth. This also maintains the elasticity of the seam. (When I sew without taking pictures, I hold one hand behind the presser foot and one in front to pull the fabric) 
Finished seam, outside and inside. Since the garment was already sewn and pressed, it was relatively easy to hit inside the edge even though I sewed on the front side. More pins can help if it's difficult, just make sure that the inner edge is below the seam line on the outside. If there are pocket bags or anything else, make sure they are also placed back up in the waistband before you sew. 
The shorts had a reinforcing seam over the pockets, so I sewed over it an extra time with a narrow zigzag stitch with short intervals between the stitches. 
The shorts were finished in time and ready for a little vacation trip 😊
If you want to add a drawstring to pants with elastic that are just a little too loose, you can proceed in exactly the same way, with the only difference being that you sew buttonholes into the waistband yourself after you've ripped it open. Mark equal distances from the center front, preferably 2-3 cm from the midpoint on each side, and sew buttonholes or punch eyelets. Voilà, the pants fit!
Good luck 😊