Great idea re: the SSID - Melbourne is actually currently rolling out free WiFi for the entire city, also most of the populated areas around town offer free wireless from nearby shops.
My only issues with free wireless hotspots is that they tend to be unencrypted which can be dangerous, however if you're in need of assistance this might be worth the risk.
*Edit: Also we have free wifi if you're near our office building or the near by train station which is an area with a lot of homeless people, from your idea around the SSID name - we'll do that here and spin it up this morning - so thanks for that!
I could still see small hotspots being a useful way of spreading knowledge about the project. You could then also make in window stickers for businesses that offer the service to also promote it.
Great idea, but -- why limit it to one site? If bandwidth is the concern, I'd still suggest making available a small suite of sites of special interest to the homeless.
As a kiwi (Christchurch) that moved to Australia, oh how I miss New Zealand's Internet, the quality and speed of Internet connectivity is a tremendous problem in Australia, I remember when I moved from NZ 4 or so years ago I went from having 150/40Mbit fibre to realising that all I could get in my brand new apartment in Melbourne was 6-12Mbit of unreliable ADSL if I was lucky, i hadn't had ADSL speeds since perhaps back in my high school days. On the flip side it's fantastic to see so many adorable mobile providers compared to NZ, while the speeds may not be great it's more than enough for people in need to find the services they need.
Remember that people often have phones but are out of data/minutes when they need help the most.