I've been stepping on LEGO parts for the better part of 20 years. Oh yes, you heard that right.
20 YEARS!!!!!
After looking online at some creative (and intricate!) sewing solutions, I ultimately decided to stick to the K.I.S.S. method.
Simplicity really IS best.
A simple circle cut from cotton twill (bought cheaply from a remnant bin), edged with bias binding to form a casing (a repurposed cotton curtain bought at the charity shop) was all it took.
We scrounged some cording from the garage and threaded it through to form a drawstring (Reece's contribution to the project). The toggle was harvested off of a casted off garment ages ago and was put back into useful service on this project.
The mat forms a great play surface as it's a nice solid colour, making it easy to spot all those tiny little parts... but the magic truly happens when it's time to clean up!
A 30 second fix - simply gather up the cording (drawstring fashion), secure the toggle to keep it closed, and store the closed bag (now sack) lickety split!
What I would have given 15 years ago to have something like this....
Edited to add:
********PLEASE NOTE THAT THIS DRAWSTRING IS A STRANGULATION HAZARD FOR YOUNG CHILDREN. IF YOU CHOOSE TO MAKE THIS PROJECT, SUPERVISE YOUR CHILDREN CLOSELY WHEN THEY USE IT*******