I'm hoping that the child is already safe and sound as the answer to this question is quite difficult to provide for numerous reasons. If the child is conscious and is able to communicate and the bore hole is large enough, a simple sling on the end of a rope may be the best bet. If the child is unable to communicate for any reason, you need to get a camera down there to asses the situation as well as some tubing to supply fresh air. The orientation of the child in the bore hole will make a difference in how you proceed. There have been a few devices invented in recent years to rescue children from wells, but as with the sling, the child needs to be conscious and able to follow instructions. There were a couple devices that were invented in India a few years back that does not require interaction from the victim but they have complicated mechanisms and require fabrication and may not be available in your area. Getting something like a cut pipe that is slightly smaller than the bore hole that can be inserted between the victim and the bore hole and then squeezed enough to prevent any downward slipping is a fairly simple method but again requires some fabrication and materials that may not be easily accessible in all areas.

Some of the techniques illustrated in this video may be helpful. https://www.youtube.com/watch?v=5iI74LQYbqk

I wish you all the best!