When you import an MP3 audiobook it appears in iTunes (and your iPhone) under music because iTunes assumes anything MP3 must be music. Here's how to move it to the Audiobooks section.
When you borrow audiobooks from your library, there are usually two options: MP3 and WMA. I will go for the WMA audiobook if there is an iPod/Apple license because WMA are imported as audiobooks and MP3s are imported as music, but sometimes the only option is MP3.
The reason why I don't like this is because the iPhone doesn't allow you to play music at double speed and puts the files on a different tab so you have to remember where to go to find each book on your phone. I always listen to books at double speed and want all my books under the same tab….so I need to move it.
2. Click on the Options tab and under "Media Kind" select "Audiobook" then click OK.
3. Go to your phone. Open the Music app. Click on More, then Audiobooks. Your book should now appear. If it doesn't then try going back to iTunes and doing a sync.