There are at least two variables at play with the Lightning Network that could affect the price of Bitcoin.

The first is perceived value. Bitcoin with the lightning network activated will have more intrinsic value because the block chain will have been upgraded from being a narrow and serial network, to an infinitely parallel Network.
The second has to do with how a Lightning Network works. In order to use the Lightning Network, two parties first create a peer-to-peer payment channel that is initiated on the block chain.Any Bitcoin put into the payment channel is now locked to that channel until it is sent back to the block chain to be finalized.