How do you set what thumbnail always appears whenever you post your homepage link in social media?
That all depends on what you designated in your code. And when you last refreshed thumbnails on your social media page.
In a nutshell, you use Open Graph Meta tags inside the <head> of your HTML document something like this for Facebook:
<meta property="og:title" content="How to Become a Web Expert in 8 Days" />
<meta property="og:description" content="Getting from web newbie to web pro in 8 simple lessons." />
<meta property="og:image" content="https://example.com/social/fb-how-to-become-a-web-expert.png" />
Just to expand on that.
You also need to have the right image size as well. Linkedin especially if it sees the image to small in the Open Graph Meta will just go off and pick the next best image it thinks it should use - And is very BAD at that!
Other issues mainly around facebook will be how pixelated and blured if too small.
Best sort of base is - 1200 pixels x 627 pixels (1.91/1 ratio)
This is why often it will be either a thumbnail generated from an image through some form of CMS setup or you provide a seperate image over just one picked from content or an intro blog post etc.
If you want tweets to work well you want to add aditional twitter card meta tags as well.