It sounds like you have two separate projects:
- Capture and archive the audio.
- Share the captured audio.
For the first, you'll want to be very careful to avoid loss in audio quality. The approach would likely depend on how these recordable books work. Ideally, you could extract the digital data directly from the media it's stored on. If it's in a proprietary format or otherwise inaccessible, the next best thing is to capture it directly from the audio output of the book, without any intervening microphone. Then be sure to archive it in a lossless audio format such as FLAC. MP3 is a lossy compression algorithm, that causes the filesize to be smaller, but also results in a loss of audio quality. In this day of incredibly cheap data storage, it's pointless to sacrifice audio quality for the sake of saving a couple hundred megabytes of disk space.
As for sharing, once you have an audio file you could simply share it in that form. However, you might prefer to have a dedicated physical device for this purpose. That is where the Arduino could indeed be useful. You can get MP3 player modules (I think the popular one is the DFMini) to connect to the Arduino to allow it to play MP3 files. You can then generate MP3 files from your archival lossless FLAC files and use those on the Arduino-based playback devices.
So the tricky part is the archival, but that's worth making some effort to do right when it comes to irreplaceable heirloom data. Once you have a good archive of the data, the rest is easy enough. It should be a fun and meaningful project!