"Small, serene, and lightweight" is a tough order. Most light little boats suffer terribly when the wind kicks up -- they have a narrow window of sailability, mebbe fifteen knots. It seems you are always either drifting at snail's pace or hanging on like grim death.
You could get a chunky little dink & rig it short, just tool around. There are a few under-canvassed 'fambily daysailors," like the Snipe or
American 14.6. The
Precision 15is a serious, serene, stable, six-hundred pound bundle of love.
If old-fashioned appeals, a gaff-riffed
catboat may be your ride. They have soul to spare, and are forgiving at the expense of anything resembling performance. Google "family daysailer" for the usual embarrassment of riches.
Oh-so-agree with USPirate's suggestion of a Compac 16. Sailingtexas.com has a video of one sailing around. You can see the wind is blowing pretty good, and the boat is moving nicely -- but barely heeling or pounding. I'm SO envious -- singlehanding w/out hiking straps! What a concept.