Statistical study-wise in the 1865 census, there are only two women living in Orkedal with the name Fredrikke (and all logical spelling variants) Larsd.:
Sokn Gard Personnr. Førenamn Etternamn Fam. stilling Sivilstand Alder Kjønn Fødestad 4140 Ørkedal Togstad 9 Fredrikka Larsd. hans Kone g 62 k Ørkedals Prgj. 5961 Ørkedal Andølsøien 16 Fredrikka Larsd. hans Kone g 31 k Ørkedals Prgj.
It is exceedingly unlikely that the 62 year old woman would still have been in the baby business in 1861 and her husband is not Ole/ Ola Olson. (And, of course, the 31 year old woman is married to an Ole Olson and has a son born in 1861.)
There appear to be no persons named Olden / Oldin living in Orkedalen in the 1865 census and no persons named Olden/ Oldin born in Orkedalen living elsewhere in Norway found in the 1865 census.
Frankly, given the information on the death record in New Zealand and these facts in Norway, make it exceedinly unlikely that Olden Olson could be other than Ole Olsen age 5 on Andølsøien in 1865.
Olden Olsen who died in New Zealand 1936 was from "Orkkedalen" Orkdal in Norway. The only parents Fredrikka Larsdr./Ole Olsen in the whole Norway with a son was Ole born 1861 in Orkdal.
Perhaps info on what happened to farm Andølsøen between 1875 and 1900, called Annølsøien in 1900 exist in the following links. Andølsøen/Annølsøien was a sub farm under farm nr 1.
Here are many digitized records and pages from the Real Estate Registry from Orkdal link