My domain name is coming up for renewal (it will expire next month), should I? ?

I'd like to start with a "free" website using the same domain name I have now. Would it be better to let my domain name simply expire and get it again when I get the free website or what? I'm lost and confused.

you will have to renew it. It will be the best idea.

Try and get a few small businesses to advertise on your site

that will pay for it

You should keep the name. They're cheap. You can register a name (remember, registering is separate from hosting) for a mere $5 to $8 per year. You can afford that, no? Like, a paperback novel.

Once you drop it, it may be lost and gone forever. Will you regret it? Is it unique enough that it'll be hard for you to think up one you like as well?

(If you're hosting with the registrar, then move the registration to a cheap registrar and drop the host: you'll be glad you did.)

Then, like you said, get a cheap or free web host and use the control panel on your registrar to point the A records and the CNAME records to the cheap or free web host. (When you get the cheap or free web host, make sure, before you choose one, that they will let you place their IP address as a target for A and CNAME records...some freebies don't.)