Alison Andrus (North Dakota Project, among others) has sent the following update info:
I followed up on the BLM record for Lorenzo Dodge.
That is definitely the right guy. He purchased
the land at 131N, 51W, Sec. 21. His brother,
Artimus W. Dodge purchased Sec. 29 in the same
township (Liberty Township) a month later
on Jun. 20, 1883. Since these were cash
purchases, they were not required to live
on the land. This seems to be the case
since they are not listed on the 1885
Census for Dakota Territory.
There is a notice of foreclosure with an
ad for a mortgage sale in the local newspaper
for Lorenzo Dodge's land that can be seen
here (second paragraph down on the far right); https://chroniclingamerica.loc.gov/lccn/sn84024779/1891-04-30/ed-1/seq-8/#date1=1789&index=7&rows=20&words=Dodge+Lorenzo&searchType=basic&sequence=0&state=&date2=1963&proxtext=lorenzo+dodge&y=0&x=0&dateFilterType=yearRange&page=1
Evidently, Lorenzo was making payments before
the foreclosure in 1891 or legal action would
have been taken sooner. Since he doesn't
appear to have actually lived there, he was
probably using it as an investment or renting it out.
The 1897 plat map for Liberty Township, Richland
County, shows both those sections owned by other families.
The Richland County history makes no mention of
anyone with the surname Dodge. I checked the
death index for North Dakota and he isn't listed
unless he died before the index was started in 1893.
He doesn't have a grave marker in ND, SD or MN.
I followed up on the L.A. Dodge that was found
in MN, but that proved to be another man.
No leads on this guy. I can see why he
is a family mystery.