Hy g_2_the_izzo,
i do teach private lessons to some childs too and my first suggestion is to buy an economic classical nylon string guitar ( 60-80$ ).
I suggest this "cheap" guitar cause before to develop a good sound it takes a lot ( a good player play good also with a cheap instrument ).
I prefer to suggest the nylon strings guitar to my fresh students cause in this way it would be easier to approach the instrument without suffer too much with steel strings. I make my students to discover the different way of playing the classical guitar, with the fingers and the pick too. Sure the neck of a classic one is different from an electric but if the hands of the child are big enough it's a good idea to start "easy" , especially for the pressure that a fresh guitar player will apply to the strings. he will think at first to use many pressure, actually too much, but this kind of mistake could become really difficult to fix later. Cause at first it seems they need to apply a lot of pressure but when they get the right balance they will discover that it's not necessary too much pressure to make sound the note. All the excess of pressure it's a big waste of energy an dexterity. It's better to discover this without having immediatly the fingers bleeding on a steel strings guitar : ).
It depends also wich kind of music he wants to play. Some child immediatly wants an electric guitar but i always try to explain to them that an economic nylon strings is better for the start. If they will enjoy the music and the instrument they could have a better or electric one.
This is my personal humble opinion.
All the Best
Animitta