we are having a hard time finding a restaurant that opens early near Hotel Hafen Hamburg. Out hotel offers breakfast but it would cost $50 per day for two of us. We just want a cup of coffee and maybe a breakfast sandwich.
Using Tripadvisor and searching for places serving breakfast in the immediate and nearby neighbourhoods it came back with a few results.
You'll want to look through and make your choice based on your preferences. I've never been to Hamburg so cannot advise.
This cafe is just a few minutes walk from your hotel. You can look at the menu (speisen) at their website.
And I saw a Backhus at a few minutes from your hotel, next door to the Aldi supermarket (here on the map). According to their website they have a café along their bread business.
For those on the go, I've found Germany one of the easiest places to get a reasonably healthy, quick and filling breakfast. It comes in the form of a (cold) sandwich, which can be obtained on a wholewheat-y roll and is a reasonable size, not like those monsters sold in many other countries. Most places selling breakfast pastries have an array of sandwiches displayed as well. You need to be up for things like lettuce and cucumber at breakfast, though.