If the elastic still springs back and its slots are sound, replace the missing anchor button on the inside of the waistband. A waist that has become loose because the elastic has stretched out needs a different repair.
Identify the anchor point
Turn the trousers inside out. Look for old thread marks where the button came off, then compare with the other side if there is an identical adjuster. The button holds one of the slotted elastic holes. Its location should allow the elastic to sit flat and to move along the casing as the wearer adjusts it. Clip the visible end before working if it could disappear into the opening.
Choose a button that fits the elastic slots
Use the old button or a matching flat one. Try it through a sound slot on the exposed elastic before sewing. A small button that slips out when tugged will not hold the adjustment. A thick button or one that has to be forced through can stretch or tear the slot. Match the other side’s height and shape where possible.
Stitch into the waistband facing
Mark the original spot and check there is fabric strong enough to hold the thread. Sew the button to the layer the original used. Keep the elastic and the opposite face of the waistband clear of the needle unless the garment construction requires otherwise. Too much slack under the button can let it tilt out of the slot; a button pulled tight into soft cloth may be hard to use. Secure the thread and remove the clip.
Try the adjustment without the garment on
Set the elastic at one loose and one tighter position, then gently pull the waistband to see whether the button holds. Verify that both sides can still be adjusted and the elastic does not twist. Check comfort and fit on the wearer afterwards. If the original anchor fabric has ripped, repair the base before adding a button. If the elastic no longer recovers or its slots are torn, ask about replacing the adjuster rather than sewing the button through the worn elastic.