I would record the LP’s and enhance the wav-files (hiss/click removal) with Cooledit Pro.
You’ll need some sort of pre-amplifier because the signal from your LP-player is too weak. I’m using a Turntable with phono-connectors for this purpose, but any HIFI-amp with LP-input and stereo output will do.
LP -> AMP input
AMP output (cinch stereo; red/white) -> Soundcard Line-input
After recording and enhancing the tracks, you can record them to minidisk using either the stereo output from your soundboard or an optical output (if both your soundcard and minidiscrecorder support the optical in/output)