Something like Audacity should work fine. It sounds like the problem is somewhere in your setup, not with the software. Make sure you have a good, tight connection with the wire. If you're using the headphone jack for recording, the recorded sound will be very low in volume because there is nothing to amplify the signal. In that case, you'd have to use some kind of program, like Audacity, to raise the volume of the recorded sound. It's better to plug into a mixer 1st, or some kind of amplifier, & then plug into your recording computer from there.
Don't overlook the most simple of things: make sure the volume on the Gameboy is raised all the way up.
And mentioning that, make sure whatever volume you have the knob at isn't cutting out. These devices are old, & the volume potentiometer on many of them have dead spots & areas that cause static. Feel free to look here concerning that issue:
http://chipmusic.org/forums/topic/12200 … is-repair/
Echos??? Make sure that any microphone device on your computer is turned off while you're recording, just in case. Check for that in volume settings & audio settings.