How important is it for your website to have a unique domain when applying for academic jobs?
I don't think there is any real downside to the weebly.com domain name.

There isn't really an expectation that you have a personal website in the first place in academia.  It is "something extra", and the fact that you have it hosted on a free service doesn't seem like a downside to me.

Note, however, that if you do want to register a personal domain, you can do it for about $10 or so a year, much less than Weebly charges.  You can't map a Weebly site to a domain without a paid plan, but you could still have the domain redirect there.  Also, there are other free hosting services (e.g. Google Sites) that allow mapping the domain without extra charges beyond the domain registration.

Also note the suggestion of Nate Eldredge and others that you could use your university web page as an alternative that is free and looks more "professional".  Also, it may help your work be more discoverable.  It does have the downside of not coming with you when you change institutions, though.