How do you share audiobooks on family sharing?

Home sharing involves using the iTunes library on your computer. If you open iTunes on your Mac you can see Audiobooks on the left sidebar under Library:

If you don't see the left sidebar, go to Vies>Show Sidebar. If you can see the left sidebar but don't see Audiobooks, go to iTunes>Preferences and check Audiobooks in the Show section at the top of the General tab.

If you don't see your Audiobooks in your library after selecting Audiobooks on the left sidebar, connect the device that has the audiobook to your computer, then go to File>Devices>Transfer Purchases to transfer a copy to your iTunes library. Once it's in your library, your can transfer a copy to other computers on your network using Home Sharing (see http://macmost.com/sharing-ios-apps-with-family.html).

Sorry, "aby" was a typo that should have been "any".

First some background. iTunes has some anti-piracy measures built in that prevent a device that already has purchased media on it from syncing with a new computer. When you attempt to do this, you will be warned that iTunes will erase any purchased media currently on the device and replace it with the media in the iTunes library you are syncing with. While you could theoretically just download everything he has on his device now again, you would lose any app data (such as game progress) in the process of having his apps deleted then downloaded again. The assumption is that any iOS device will be synced with iTunes prior to adding purchased media to establish a syncing relationship between it and they computer it will be synced with.

In order to avoid this problem, prior to syncing with your computer the first time you will need to perform the steps outlined in this user tip from wjosten: Syncing to a "New" Computer or replacing a "crashed" Hard Drive (while this refers to iPhones, it's the same for any iOS device). Step 2 (and the corresponding reference to contacts and calendars in step 9) will only be necessary if he was syncing contacts or calendars with a different computer using iTunes. Assuming he wasn't (such as if he was syncing them with iCloud), you can skip this step. Step 4 refers to authorizing your computer for any Apple IDs used to purchased media on his iPod touch. If he was already using your iTunes store ID for his purchases you can skip this step too as your computer is already authorized for your iTunes store ID. At step 5, transfer purchases is now in File>Devices. Finally, since he hasn't synced any photos to his iPod from a computer, or added any music from CDs by syncing with iTunes, you won't have to worry about the reference to extracting these using 3rd party software.

What this will do is transfer his purchased media to your iTunes library, and allow you to sync his device with your library without any data loss (such as game progress). After doing this you can select your audiobook to sync to his device without any issues.

iTunes will keep separate sync settings for each device but you will want to review your sync settings prior to your next sync to avoid inadvertently getting his apps, etc., on your device. You will also probably want to uncheck "Automatically install new apps" at the bottom of the apps list in your iTunes sync settings as shown in the image below. This will prevent new apps and other media added to your library from his device when he syncs from automatically being added to your device the next time you sync (and vice versa).

You cannot download previously purchased audiobooks (or ringtones). As a result, they cannot be shared with others using Family Sharing.

If you want to authorize another family member's computer for your iTunes store account, you can transfer a copy of the audiobook from your iTunes library to theirs using Home Sharing. Then they could sync it to their device(s) using iTunes. Just be aware that your are limited to authorizing 10 devices for your iTunes store account, of which 5 can be computers. See:

iTunes for Mac: How to copy purchases between authorized computers

iTunes for Windows: How to copy purchases between authorized computers

It wouldn't matter when they were purchased. Family Sharing allows members of your family group to download purchased media from each other's iTunes store account by accessing them from their Purchased list. This includes music, movies, TV shows, books and apps (see Which purchased content can I share using Family Sharing). Audiobooks will not appear on your purchased list because they are excluded from iTunes in the Cloud.

You can still share them, but you would need transfer a copy to your other family member's iTunes libraries using Home Sharing so they could sync them to their device(s). However, as mentioned, you could only transfer a copy to 4 other computers as you cannot authorize more than 5 computers for your iTunes store ID.

What version of iTunes are you running (iTunes>About iTunes)?

Also, did you transfer the audiobook to your iTunes library by connecting the device that has it, opening iTunes, then going to File>Devices>Transfer Purchases?

You're probably using a PC rather than a Mac. On a PC, you'll find them in the Books section.

Click on Books under Library on the left sidebar, then click on the Audiobooks tap at the top to the right and see if you see your audiobook(s) now.

In the meantime perhaps you can answer two questions: does he have aby music on his device that was not purchased from the iTunes Store? Also, does he have any photos on his device that were synced to his device from a computer using iTunes?

Did you open iTunes and go to iTunes>Preferences>General and check Audio Books at the top under Show?