Can a mangaka continue another mangaka's manga? There are plenty of manga that never had an ending for many reasons. For example, High School of the Dead's mangaka, Daisuke Satou, has passed away. Some mangaka also cannot continue their work because of health problems, as well as other reasons. Say the original mangaka either prepared or did not prepare an ending himself/herself, can another mangaka continue their work? Has this ever happened before?
Yes, it can. It is usually up to the publisher if they want to pursue continuing a series, depending on who owns the rights (family member tend to have a say in it as well). 

Note that for HotD, the plot creator was known to be a slow writer to begin with (2 years between vol. 6 and 7), and the manga was on hiatus for quite a while before his death (2011). While his death was the nail in the coffin, I don't think anyone really expected the series to ever be completed. The artist moved on to create Triage X, which basically was written as a replacement for HotD.

Below are some examples (though I'm sure there are more).


  Cases where the author died, and his team took over:
  Crayon Shin-chan
  
  Cases when the author changed, even when the previous is still alive (though it is technically a new series or spin-off):
  Lupin the 3rd
  Saint Seiya
  
  Cases where the artist died, but the original author was still alive:
  Catapillar (They replaced the artist with another one)
  
  Cases where they had someone else fill in for the dead author:
  Familiar of Zero