http://www.youcanquiltthis.com/info/stitchery-hand-embroidery/FREE-Stitching-Essentials.htmHere is one site for a free pattern with instructions. YOu can also check at the sister company: youcanmakethis.com for more free project pages.
You don't need to be an artist to "draw" a pattern. Write or print your name on fabric. Have your child draw a simple pattern on fabric with pencil (tape sides of fabric down or iron it to the waxy side of freezer paper at stabilize it while they draw then pull the paper off before you stitch), You can also tape down a page from a children's coloring book, tape fabric over it and trace the pattern onto fabric.
Good luck and congratulations on trying something new.