If your plan allows your own domain branding then your site should come up as searchable on Google. In the search results you have (a) the search link wording and (b) the snippet text. The search link wording is based on your main page "<title>" field (which by default is set to your organisation name) and the snippet text is from the first text in your site "Home" page (most likely a snippet such as the "Home-Benefit-Left-1 Snippet").
To set something other than your organisation name as the "<title>" text, or to change the title text at any time, go to "Content" > "Branding" and update the "Web page <Title>" field. If you delete the field completely it will default back to your Enlight organisation name.
Google search results example (not Enlight!). The URL at the top is your site, the larger purple hyperlinked text link is from your main page title, and the text that follows is the first paragraph or so from your web site:
Editing the title:
Title showing in a web browser.
Editing the snippet: