networking - Extending WiFi range - what do I need and in what configuration?

The setup would be as the diagram below.

The issue is that when in Room A, the WiFi signal from the router/modem in Room B is flaky, causing the devices in Room A to drop the connection all the time.

I can easily add another 3-port Powerline networking plug in Room A (I have them elsewhere) to get wired connections into the Wii/XBox/Sky TV box.

My question: I want to get the two WiFi devices in Room A connected via the 'mystery box', which will allow a strong signal since it is in the room, and which will use the (effectively, via Powerline) wired LAN connection to the modem/router in Room B, and not just pick up the existing WiFi signal from Room B,

So do I need something in repeater or access point mode for my mystery box, and can it be made to work over a wired connection?

Answer

Looks to me like your mystery box needs to be a wireless access point.

If you'd like to roam between both WAPs (one in room B and one in room A) then set identical SSID and security settings. I would give this a try and see if it works well with your wireless devices. Wireless roaming is left up to the client device/network adapter, so your mileage may vary between machines (my Galaxy S3 roams like a boss; the wife's ancient Mac book not so much.)

If roaming doesn't work well for you, I would simply change the SSID and ignore/shut off the WiFi in room B. It depends greatly on the equipment, but I have yet to see a decent consumer repeater, and since you can run a line to your mystery box I see no reason to use a repeater.