There are many thoughts on how old a child should be before you trust them with a needle, but teaching the concept can begin during toddlerhood.
As soon as a child starts grasping things with their thumb and pointer finger, introduce simple lacing toys. Don't expect to be able to explain how to do it and have lacing pros. Instead, let them fumble through it and get a feel for cause and effect.
As they get older, you can upcycle some great things into lacing tools. Cut shapes.out of old maps, laminate them, punch holes around the edges, and string an old shoelace through the holes. Cut a cereal box up and punch holes around the edges. Punch holes in old greeting cards. The possibilities are limitless.
Once they become lacing pros, it's time to move up to yarn, yarn needles, and plastic mesh. Again, don't expect a masterpiece. The process is the important part, not the product.
Before you know it, they'll be ready for the real thing. There's no need for expensive craft kits. Just get an embroidery hoop small enough for them to maneuver, some embroidery floss, and a tapestry needle. Cut a piece from an old tee shirt just big enough to fit in the hoop. You will probably have to thread it the first 100 times, but let them try until they ask you to do it. If they feel discouraged, remind them that it takes a lot of practice. When you thread the needle, tie a knot around the eye so the thread doesn't come off a million times. Tie a knot in the end of the thread, and show them how to pull it through the fabric. Again, this is a process-not-product activity.
It's never too late to get started, and you'll be giving them a valuable gift that goes far beyond poking a needle through an old piece of tee shirt.
Trust me, you're gonna love it!