A "free domains" registrar won't be a charity - nor are registrars, in general, non profit organisations. There will be differences - between a "free domain" - and a properly purchased domain, from a full service, reliable registrar. Features cost money - and even "free domains" have to be paid for.
Google "free domains for blogs", and see what you get. What they offer may not be what you need, for Blogger custom domain publishing.
There are various features, that you may not get, with a "free domain".
- Legal ownership of a top level domain.
- DNS hosting for a top level domain.
- Properly designed and maintained server infrastructure.
- Customer support staff, to help you the new "owner".
Legal ownership of a top level domain.
The earliest versions of the "free domains" services, ".co.cc", and later ".co.vu", were blatant fraud. Later "free domains" would register a top level domain - but in the name of the domain "registrar".
With a normal top level domain, you, the customer, "own" the domain in question - for the period of the paid registration. With a "free domain" you will have no registration - just use of a subdomain / domain, under the limited conditions granted by the registrar.
If you are going to go to the trouble of publishing a Blogger blog to a non BlogSpot URL, you need a genuine top level domain. A subdomain will not provide the same results.
DNS hosting for a top level domain.
Some "free domain" hosts provide name registration, for an actual top level domain. In this case, you, the new owner, must setup the DNS hosting.
If you have not, to date, setup DNS hosting, beware - it's not an ideal project for someone who has not setup a custom domain previously. Nor is it as simple as purchasing a domain from a reliable registrar, and getting assistance from "Customer Support".
You can possibly use ClouDNS (also free) for a domain host - if the "free domain" works with ClouDNS.
Properly designed and maintained server infrastructure.
The best full service registrars maintain a world wide network of domain servers. This makes it likely that one server - which hosts your domain DNS - will be close to your readers. "Free domains" registrars may not provide this infrastructure.
If you know anything about IntoDNS and similar DNS checking services, you may run an IntoDNS evaluation of your domain. What you see, in a comparative evaluation of a "free domain" may not be pleasant.
This feature may or may not be important to you, immediately. If relevant, you will probably figure this out, over time.
Customer support staff, to help you the new "owner".
Live, one on one domain support, if provided by a "free domains" registrar, will be part of their "premium service" package. "Premium service" is not going to be free.
If you're able to setup the DNS addressing correctly - and if the registrar sets up the domain correctly - you may not need this. But of you do need it, and have to suffer without it, you will regret your choice. Only time will tell.
If you know what you're doing, you can probably make a "free" domain work.
For someone who has setup a custom domain already - or an experienced DNS technician - a "free domain" might be fun to setup, and to use. The concept is valid - if you are choosy.
If you are a first time blog owner, who simply wants a non BlogSpot URL for your shiny blog, plan to pay for the domain. Don't start, on the cheap - and expect to get a working domain, without a lot of effort - and substantial technical background.