What does "famously absent" mean?

Solution 1:

“Famously absent” means that that absence is particularly well-remembered, the reason being that he was absent on the night of Abraham Lincoln’s assassination, and as this is rather suspicious, this absence was rather notable. This sort of usage is common, such as:

James’ famous speech (if his speech was particularly well-remembered for some reason or other)

Coles’ famous advertisment.

Basically, Lamon might have been absent on another night, but that wouldn’t have been really important. But since it was the night of Lincoln’s death, it stood out very well.

Solution 2:

Famously is an adverb that is used to indicate that a fact is widely known.

They have famously reclusive lifestyles.