As you have desribed the situation, it does sound like the neon graft is rotting. I can not tell you why, but I have seen it on some of the plants they sell at the store. If you do decide to save the bottom piece, take a sharp knife or pruning shears and cut it where the two pieces meet (where they are grafted together).

I believe the cutting on the bottom is hylocereus unadata (sp?) also know as dragon fruit, hopefully it will grow for you and your daughter. Then she can learn about tropical fruits and cacti at the same time. :)

I have a similar cactus....bought it for 99¢ at WalMart to put in a cactus garden. It is just like the yellow one seen here:

http://www.hisupplier.com/userImages/sunshi/sunshi$95131815.jpg

I went to water my cacti today (haven't for quite some time) and noticed that the base cactus appears to be rotting. It has yellowed some in color and seems to be getting softer. I have this at work...no natural light but located approx. 12" under a flor. light. Any suggestions on how to save the top cactus??