Currently (Dec 2017), when you enable SSL on a Weebly site, the certificate is valid only for the www version of your domain (ie https://www.your-domain.com). It is not valid for the non-www version (ie https://your-domain.com).

So, even though weebly will serve both versions of your domain, the non-www one is declared as insecure by browsers, scaring away your site visitors.

I have discussed this with Weebly and they suggested I share my idea of fixing this situation here on the Weebly Community!

I noticed that the certificate for my site was issued by LetsEncrypt - I presume that is the case for all users' sites. LetsEncrypt's website states that they will be introducing wildcard SSL certificates from Jan 2018 (next month). A wildcard certificate covers all subdomains of https://your-domain.com such as https://www.your-domain.com, https://blog.your-domain.com, https://shop.your-domain.com, https://mail.your-domain.com, etc.

So, once this is available from LetsEncrypt, it should be a relatively easy fix for Weebly to request and install wildcard certificates, then browsers will be happy with any part of your site no matter what address your visitors entered to get there.