iPad Mini and WiFi Connection
With iOS6, the WiFi will drop when the iPad is put to sleep (via power button or closing the smart cover).  When you wake it up, it should automatically reconnect to the WiFi.  However depending on the WiFi environment (many WiFi networks to choose from, a slow/overloaded WiFi router, etc), this could take a few seconds.

If it isn't automatically reconnecting, go into the Settings>WiFi screen and choose your WiFi network.  If it is already checked, click the > on the right and choose to forget that network.  Then you can go back and set it again.