Extenders will decrease your speed and network bandwidth.
The best way to go about this is to place an Wireless access point aka WAP (Commercial Grade) at the highest point in the house dead center. This is a problem I encounter everyday a work. The gear we sell is designed to cover min 2000 sq ft. Up to 6000 Sq ft per WAP usually I need 2 -3 to get perfect coverage in my client's houses. With all this being said...you will need a budget of about $300-$400 to deploy the network it sounds like you are looking for. BTW, you will need to find away to purchase the commercial gear and setup/install it. This would be the proper way to do the job.
The realistic solution...
http://www.netgear.com/home/products.../XWNB5201.aspx These dont always work, it depends on how your house is wired. So make sure you can return them from where ever you purchase them. Plug the send unit by your router, connect a network cable from one of your out ports on the router to the send unit. Plug the receive unit in your relative's room and you should be good to go. If it doesn't work try a few of the other rooms around the target room before you give up. Again it does depend on the electrical circuits of the house. Netgear's stuff is pretty easy to setup, just read the directions once over before setup and take your time walking through it, there are a few things that are easy to miss and will cause an installation to fail.
I have not used the wireless version of these, but I have used them in a pinch and I have been very happy with the results. Shoot me a PM if you have more questions or setup questions, I will be happy to guide you though the best I can.
- Mike