@UnaiB Thank you for the shout-out.
I'll go the eastern route, as people have already mentioned the west (most likely alternate location). I haven't seen any concrete dates on the Navajo migration, but we'll use 1000 CE (a glottochronological date for the split between Navajo and Apache).
I could see the Navajo going into the Mississippi Basin, traveling through the northern Great Plains along the Missouri River. They could end up anywhere along the river, but for dramatic purposes, by around 1300 CE, they reach the crossroads of three great rivers, home to the heart of the Mississippian Culture, within it the dying city of Cahokia.
Do they go to war with them? Assimilate? Replicate their mounds and lifestyle? Hard to tell, but in the Southwest they did a bit of all the above. Navajo moundbuilders would be cool. Historians ITTL will probably think the *Navajo caused the fall of Cahokia.
Could they break the Mississippian cycle of periodic collapse and establish a stable, more permanent, way of life? A *Navajo leader could decide to conquer the Mississippians and their constantly fluctuating chiefdoms under a more centralized rule, with a *Navajo elite. If they do, they could maybe live to see the Europeans, and some colonization butterflies ensue.
ITTL, they probably assimilate the religious aspects of the Mississippians, as they did the Puebloans IOTL. More thunderbird and underwater panthers in their mythology than in OTL.
In conclusion, my conservative prediction is that we probably see an anomaly, an Athabaskan speaking people in the heart of the Mississippi, for linguists and anthropologists to argue about. But it's a pretty open scenario. Could a migration and its accompanying social pressures lead to new innovations and cultures? Who knows.