Save email attachments from MS Outlook 2016 to SharePoint site

With SharePoint Online and Office/Outlook 2016 you have three options:

1. Use out of the box capabilities, although limited in a few areas and key support at this time for OneDrive for Business only
2. WebDAV which personally I don't like 
3. Third Party Solutions, and on this there a few and they come in two flavours
   • Traditional COM / vsto add-in (desktop client installed on each machine)
   • Office Store Add-in (app) which can work in Outlook 2013, 2016 but also on the web browser and macOS (very nice):

Note: If using the

•    Traditional COM add-in, most third party solutions will save as an msg file (more proprietary to Outlook, not exchange)
•    Office store add-in, you will be saving an email in .eml format (can still open in Outlook fine) 

You didn't mention it in your question but most business would like to capture the email attributes when saving the email.  This pretty much removes option 1 and option 2.  

Further to the above, you will find both paid and free offering from third parties... just google save emails to SharePoint... but hopefully the above will save some wondering...

I'm associated with OnePlace Solutions which has both a http://www.oneplacesolutions.com offering to do the above... but as I say we are one of a few in the market so look around and work through which best meets your business requirements.

James