Based on generality of this question, I think, that you'll need to setup your own HTTPS proxy on some server online. Do the following steps:
If you simply download remote site content via file_get_contents or similiar, you can still have insecure links to content. You'll have to find them with regex and also replace. Images are hard to solve, but Ï found workaround here: http://foundationphp.com/tutorials/image_proxy.php
Note: While this solution may have worked in some browsers when it was written in 2014, it no longer works. Navigating or redirecting to an HTTP URL in an
iframeembedded in an HTTPS page is not permitted by modern browsers, even if the frame started out with an HTTPS URL.
The best solution I created is to simply use google as the ssl proxy...
https://www.google.com/search?q=%http://yourhttpsite.com&btnI=Im+Feeling+Lucky
Tested and works in firefox.
Other Methods:
Use a Third party such as embed.ly (but it it really only good for well known http APIs).
Create your own redirect script on an https page you control (a simple javascript redirect on a relative linked page should do the trick. Something like: (you can use any langauge/method)
https://example.com That has a iframe linking to...
https://example.com/utilities/redirect.html Which has a simple js redirect script like...
document.location.href ="http://thenonsslsite.com";
Alternatively, you could add an RSS feed or write some reader/parser to read the http site and display it within your https site.
You could/should also recommend to the http site owner that they create an ssl connection. If for no other reason than it increases seo.
Unless you can get the http site owner to create an ssl certificate, the most secure and permanent solution would be to create an RSS feed grabing the content you need (presumably you are not actually 'doing' anything on the http site -that is to say not logging in to any system).
The real issue is that having http elements inside a https site represents a security issue. There are no completely kosher ways around this security risk so the above are just current work arounds.
Note, that you can disable this security measure in most browsers (yourself, not for others). Also note that these 'hacks' may become obsolete over time.