Hi Adam

Thanks for posting in the Lucidchart Community! Four connection points, which are centered along each edge of an image, will appear by default.  To connect a line to the image apart from these points, simply move your cursor over the outline of the image until a small dot appears at on the border. From there, you will be able to drag a line to another point on the canvas.

I’ve demonstrated this in the GIF above. Hope this helps! Please let me know if you have any other questions. 

This is not helpful as I'm pretty sure that he was referring to connection points as they are created and used in Visio. (i.e. Not limited to the edge of a shape.)

I have the same/similar request.  I am drawing network diagrams and the lines I draw represent cables that then connect to a specific physical port on the network device shape/object.  I would like to add connection points so that my lines/cables connect to a specific port - Like 1/1 1/3 1/5 etc...  If or when I move the object I don't want the lines/cables to move to a different point or a different edge or a different anything.  
So how can I add a connection point to an existing shape/object.

I've found requests for this feature since at least 2015.

Hi Brett and Cory, thanks for the additional context. Unfortunately, custom connection points anywhere on an object isn’t currently supported in Lucid, but we’re very interested in your feedback and committed to continually improving our products. Please first search the Product Feedback section of this community (with the filter Topic Type = Idea) to see if it’s already been submitted. If so, please add any additional details you’d like and upvote the request - this consolidation helps to refine feedback and properly capture the popularity of the request.

If no one has submitted this idea yet, please create one of your own and be sure to include details about your use case or what you’d like to see in this experience. This will also allow other users with similar requests to discover and upvote it, then add details of their own.

Finally, for more information on how Lucid manages feedback via this community, take a look at this post:

Here is a work around for this limitation:

1. Draw a small rectangle over the image in the position you want the ‘connection point’.
2. Snap the line end to this rectangle.
3. Group the rectangle and image together so that the ‘connection point’ moves with the image.
4. You may need to use the “Arrange > Send to front” command after the group command.
5. Change the rectangle to have no fill and no line color i.e. make it invisible.

Visio’s connection point functionality would be easer than this.

Agreed; custom connection points within an image is easily attainable in Visio and Draw.io.