Before even letting the child pick up the bow, educate him a little on the art of violin playing. Include some history and a few fun facts, as well as what types of venues regularly feature violinists. A student who understands why he is learning the instrument is far more likely to be diligent and attentive during lessons. Students who are being forced to be at the lessons will be absent-minded and will have little fun at all.
Keep lessons brief. A child's attention span is very short, so she will rarely be able to keep her focus on learning the violin for too long. Lessons over an hour can begin to get tedious and unproductive for the child, who will likely just be counting down the minutes to when she can go home. Do not try to fit five musical pieces into a single lesson; instead, spend each lesson trying to perfect the finer points of one or two pieces to maintain the child's attention.
Before handing the child the bow when teaching him a new piece, have him place his fingers in the positions where they will be to execute the specific notes. This will give him a feel for where his hands will be moving during the piece, which will make the motions needed for the bow much easier. Teach him what one hand will be doing before adding the other hand and you will find that he is far more coordinated with the two functions.
End with something fun. Pick a piece that the child may already know or enjoys playing and play along with her. This will show her better techniques for that piece by showing her how you play it, and this will allow her to finish on a positive note. The last thing the child remembers about the lesson should be something that she enjoys. This will make her more likely to look forward to the next lesson.