Buying another router and use as a "signal repeater" in bridge mode?

Solution 1:

Yes. What you want to do is called a wireless bridge, and it is possible. Not all home routers support it, so do your research ahead of time. The free DD-WRT and Tomato firmware support this and work well, so if you get a router which supports one of those, it would work. The WRT54GL works very well for DD-WRT and can do bridging.

The way bridging works is the router acts as a wireless client, meaning it connects to the existing wireless network. It doesn't matter what kind of router set up the existing wireless network. You can then connect computers to the 2nd router's ethernet ports.

DD-WRT Wireless Bridge Mode documentation

DD-WRT supported hardware

Solution 2:

I am currently using such a network. The second router has built in station mode and will connect to the first router quite easily, then you can decide how you want to route the traffic either using NAT or Bridge Mode. Bridge mode is nice because it makes your first router provide all of the IP addresses and you are all 1 big happy network. It is the way I run mine. Of course I am running advanced routers by Mikrotik. But the same thing can be done with any router that has bridging in station mode built in. I know because I have it working.