how do you turn off the vibration alert for incoming text messages on a samsung galaxy s3 mini?
On my Samsung Galaxy S3, this setting is inside the Text Messaging app, rather than as a general setting.

Open the Text Messaging app, and start the Settings menu.

About halfway down I have a block of Notification settings. 
First being Notifications,
Second being Ringtone,
Third is Vibrate.
Uncheck and you're done.
Hope your Settings Menu is similar since its from the Samsung family.