I did end up reporting this issue to Sitecore Support. Here is what they said:
I've registered this issue as a bug in current Sitecore version (8.0).
To track the future status of this bug report, please use the reference number 422229. More information about public reference numbers can be found here:
As a workaround I have only two options:
1. You can assign the sitecore\Sitecore Client Developing role to your user to be able to see the values in the Target field of that dialog;
2. You can disable the SPEAK dialog by commenting out the following line in the /Website/App_Config/Include/Sitecore.Speak.config file:
<override dialogUrl="/sitecore/shell/Applications/Dialogs/Internal%20link.aspx" with="/sitecore/client/applications/dialogs/InsertLinkViaTreeDialog" />
What I ended up doing was assign the sitecore\Sitecore Client Bucket Management role, and then things functioned properly.