So, a long running idea that has been mentioned in many alt-Reconstruction posts is the idea that the Freedmen could be granted land to homestead out West. This idea might appeal to several different groups as it would not only give the freedmen land, independence and a chance for prosperity, but would also remove them from the South (although many rich landholders would regret the loss of cheap labor, I would assume that other parts of Southron society might welcome the move after the War). There are, of course, many hurdles to this notion - but, lets assume that a Freedman Homestead Act is passed as part of reconstruction.
Since the Indian Territory joined the Confederacy...
That's a misstatement. The Five Civilized Tribes allied with the CSA, but powerful elements in each tribe declared for the Union. These were known as "Pin Indians", because they wore collar pins declaring their allegiance. (John Ross, Paramount Chief of the Cherokee, was a Pin Indian leader, even though he had been overruled by Stand Watie's Ridge Faction when the tribe chose sides.) The greater Civil War was reflected by civil wars within the tribes.
Furthermore, I'm going to suggest that this territory would, obviously, be named "Lincoln" for reasons that are obvious...
Not if Lincoln is alive, and if he has been killed, this project isn't even starting until 1869.
and that the territorial capital might be called Quincy in honor of John Quincy Adams. I would also suspect that Frederick Douglas would be appointed the first territorial governor, assuming he would want the position.
Interesting suggestion, but I doubt that even a Radical Republican administration would go that far.
So, how do you think this territory would develop over the course of the 19th and ewarly 20th century? I'm assuming it might eventually be given statehood, but that would be well in the future, assuming racial attitudes aren't radically different in the ATL than in OTL (although Republicans would certainly fight for statehood as it would give them two further Senators and at least three electoral votes).
For that last reason, it might happen in 1889-1890. (That was when the Republicans created six new states in the northwest: SD, ND, MT, WY, ID, WA.)
Also, how would having a territory dedicated to Freedmen affect African-American culture in subsequent years?
In that particular location, there's going to be a lot of conflict with the Indians. Even the Unionist Indians won't be friendly - John Ross was a slaveowner. (Famous aphorism on the grades and vectors of victimology: "When the Cherokee were driven out of Georgia on the Trail of Tears, they took their slaves with them.")
If "Lincolnia" becomes majority black and black-controlled, it will become a "Mecca" for discontented blacks elsewhere. The question arises whether it can accommodate all such migrants; it may well become a slum state, overrun with impoverished migrants reduced to indentured labor.