Good question there @texnixe. I did not really think about that option. 
If there is quote and link I think wrapping the link around all the quote text, or simply ignore it. I am not sure what is best thing. But for people that come after us here. The first option is maybe better. Also if this tag should be as versatile as possible for others to use as well.
Could you share the both examples?
I think if there is a link but no name, I’d put the link between the a tags, rather than the complete text. If there is no link and no name, well, all we have is the text.
But maybe do it like this: <blockquote>: The Block Quotation element - HTML: HyperText Markup Language | MDN, with the cite attribute? And to stick to the specs, use the cite tag for the source, anyway?
When I think of the use cases that this will be used. I think ignoring the link. Or putting it in a tags and use that as a way to ignore it. Most cases you don´t link quoted test.
Yes @texnixe great idea! Love it. I assume that it is not just me that want this. I have always wanted this tag. 
My plan is to release this tag as on my blog later with the CSS together with it. So anyone can get it. 