If a child has an electric or acoustic/electric guitar and just wants to record her playing, the simplest and most-cost effective alternative is to get a guitar/computer interface. The cord from the guitar plugs into the front of the interface, which in turn plugs into a USB port on the computer. Any of several freeware recording programs available can then record the feed.
Another direct-to-PC approach is to buy a microphone that plugs into the computer's mic port. Don't bother with headsets or the little microphones used for webcasting; you can get a good condenser mic for surprisingly little money. These have a cardioid or supercardioid reception pattern that will pick up both a voice and an acoustic guitar on one mic. Be sure to get one that is battery-powered. You may need a phone-plug to 2.5 mm adapter as well so you can plug the mic into the computer.
A good alternative to recording on PC is to use a Portable Digital Recorder (PDR). These usually have surprisingly good built-in condenser mics, and usually have external mic inputs as well. Don't buy one that's primarily designed for voice memos; instead, look for one with a 96 kHZ sampling rate and WAV file recording. These provide great results and have come down dramatically in price in recent years.
Somewhat more expensive -- but still under $100 -- are desktop or portable recording studios. These can operate either as stand-alone or computer-connected units, and they feature multiple input jacks for guitars and microphones. Best of all, you can mix the balance of these feeds as you record with the built-in mixing board. These units give young musicians hands-on experience in sound production.