If the audio is MP3, you could losslessly adjust the volume with MP3gain after demuxing, no need to re-encode.
Get
VirtualDubMod, load the AVI, click Streams -> Stream list. Select the MP3 stream (if there is one), click Demux to extract it. Maximise it with MP3Gain, then disable the existing audio stream in the AVI (by clicking Disable in VDubMod), click Add, select the adjusted MP3 file, click OK, and save. And make sure "Direct Stream Copy" is selected in the Video menu.
No time to get into more details now, just post again if you have problems.