Hi Sascha,
The reason could be that the audio sample rate in the AVI file was not 48kHz, and FCPX does not like audio that is not 48 kHz. Of course you would reply: yes but I had FCPX convert the file to ProRes, so logically that should also optimize the audio to Linear PCM 48 kHz?
Yes it should, but here’s my guess: FCPX uses the Compressor engine to optimize your video to ProRes. The standard ProRes setting in Compressor uses Audio Passthrough which means it does not alter the audio of your file, just the video.
As I prefer facts over guessing I did a test. I imported an H.264 file with AAC 44.1 kHz audio in FCPX and opened the inspector. The audio sample rate showed 44.1 kHz, as expected:

Then in the Event Browser I selected the H.264 file and I converted it to ProRes, which is the same as when you optimize the file upon import. When the conversion was finished I went into the Event > Transcoded Media folder and I opened the ProRes optimized file in QT player and pressed CMD+I to get info for the file. Guess what: the AAC audio had been converted to Linear PCM, BUT the audio sample rate still was 44.1kKHz:

This shows that when you optimize a file to ProRes FCPX does not change the audio sample rate. In most circumstances this is not an issue as FCPX does handle different audio sample rates quite well. But with your specific AVI file this could be the problem.
If you convert the audio to AIFF 48 KHz yourself this should fix the problem. But I think FCPX should do this automatically when “optimizing”, as 48 kHz is the standard sample rate for FCPX projects.
– Ronny