It's much more efficient in this case to do a bi-elliptic transfer: raise apoapsis very far out, then lower your periapsis once you are at apoapsis. Wikipedia says you could do it with about 8.8 km/s delta v. Versus 24 or so for a basic Hohman transfer (still a bit better than 30)

Sadly the bi-elliptic transfer requires two burns so you can't do it with a kick.

The minimum would be something like punting your kid to the orbit of Venus for a gravity assist that takes it to one of the outer planets where another gravity assist can push it to the edge of the solar system.
Out there, the angular momentum of the orbiting child will be very low and can be canceled out by a small thrust.
The child will then fall back into the sun. But this requires remote controlled thrusters strapped to the child. And a life support system if you want your child to actually die by burning in the sun. And then, the child will be well into their teens by the time they reach it.

The speed will be related to escape speed of sun. Based on https://en.m.wikipedia.org/wiki/Escape_velocity It will be Vte which is 16.6 km/s (or Ve 42.1). So when object at earth orbit and has lower speed than 16.6 it can't keep the orbit and will slowly fall into sun.

You could use a gravity assist from another planet