Since you are having some issues in order to keep a steady connection between the Slingbox and the BT Home Hub 3 router, we recommend you a couple of tips.
First, connect the Slingbox directly to the main router and test the Slingbox-network connections. If it works fine, the Slingbox must show a solid network light.
Secondly, connect the Slingbox on a different network. This test will allow you to know if the issue is about the Slingbox or any security software running on your LAN (local area network).
If the issue persists, we finally recommend you to check the BT Home Hub 3 router settings.