Hi Mandy, my GGmother came out to Rockhampton on the Fritz Reuter at the same time.
According to the write up in the local newspaper "Morning Bulletin" 13th October, 1877 (http://.nla.gov.au/newspaper/article/6576459)(Trove
It is mentioned that 5 children and 1 adult died on the trip over from Hamburg/Plymouth. The Captain mentions that the 1 adult died from consumption (tuberculosis). Could he have been your relative.
The passenger list can be found on Ancestry.com.au - Hamburg Passenger Lists, 1850-1934.
Hope you find it ok.