Router that extends open networks...?

You could extend the other person's network, but it would require two routers. One would be a client to their network, then a cable would connect the LAN side of that router to the WAN side of another router.

You can extend your own network using WDS, but only if your router software supports it. I have done it using my WRT54G running OpenWRT and I believe certain Airport models support this as well.

 

What you want is a wireless range booster or sometimes called a wireless repeater depending where you live

Article teaching you what you need:
http://compnetworking.about.com/od/wirelessfaqs/f/extendwifirange.htm

Example Product i use:
http://www.amazon.co.uk/Linksys-Wireless-G-Range-Expander-WRE54G/dp/B0006BA1XO

350ft is a big ask and the ones worth buying arn't cheap.

 

All current Airport models support extending a network to encompass a bigger area. As mentioned you'd use WDS mode which means that the second or third base station connects to the first base station and then re-broadcasts that. It's quite useful in a big house.

 

I've never used one of those. It is interesting that the Linksys model has reviews which state it must be used with Linksys devices.