Usually When they say on a census that they are fromHannover it usually means Hannover kingdom.There are maps of this kingdom. Browse the mailing list of Hannover kingdom at rootsweb. Boundaries will be outlined.Now Hamburg is in it.It is in lower saxony or Niedersachen.
Now the city hanover is suppose to be that with one 'n'.
But you find it used by all very differently than what I said.
Now my Heins were in Hannover kingdom, which was in Celle which is still in Luenburg stadt. Minemarried at Oldendorf by Nahrendorf church. Also some lived at Lubenand Breese. which is a skip or two away. Someof this area also belongs to Dannenberg.All of this area are several hops from the Elbe river.
If yours came from Hamburg, they may be on Aida and that may help you.