Don't trade it in for a hybrid. Hybrids are for washed up old men with bad ankles.
Kid on the back? Been there, done that. I'd put a fat tire on the back to handle the extra weight. Like a 700x35 touring tire. If the brake pads are old, replace them with something long and/or soft to get some extra stopping power.
And for the pinnacle of dorkiness, consider a mirror of some kind. Knowing what's happening behind you is just plain smart; all the more so with the youngin' aboard.
Finally, make sure your pedal system is such that you will not get stuck in the pedals at a red light. This is embarrassing when it happens to you, but with a child in back...<shiver> I know, you say you mastered the foot release years ago. But this is your old bike. Maybe you need to remember something different. Like when you go back to a standard transmission after driving an auto for years. You just forget to step on the clutch at bottom of the exit ramp.