This is expected behavior, but is meant to include the target's address as well. If your custom target does not have an address associated with it, only the name will appear. Ideally, the top of the alert message would look like this, when received by the target:
Jun 14, 2016
Sample City, ST 00000
Dear Custom Target,
This is the beginning of the message content... etc.
You can edit your target by following this path: Advocacy > Action Alerts > edit the desired alert > 4. Select Targets > click on the name of your custom target > update address fields > click Save
Connect and collaborate with fellow Blackbaud users.