Webber's Falls dates to about 1828. Charles Webber was one of the chiefs of the Western Cherokee in Arkansas in the 1820s. These Cherokee signed a treaty with the US in 1828 to give up their land in what is now Arkansas and move to Lovely's Purchase in what became known as Indian Territory.
Webber developed a saline (salt works) on the Illinois River about 10 miles north of the Arkansas River. A road led from the salt works to the falls on the Arkansas, just north of where the US Hwy 64 bridge crosses the Arkansas between the present towns of Webbers Falls and Gore. Riverboats would come to the falls to get salt which was sold under contract or on the open market as far away as New Orleans.
At the time of the Civil War, there was a thriving rural community at Webbers Falls. The Canadian District courthouse was there, a school, and a few merchants.
By 1900, Webbers Falls was a thriving town and the economic center of the surrounding communities. Therefore, the town would not have been "owned" by a family. Perhaps your ancestors had land at or near Webbers Falls. If they were Cherokee, it is conceivable (perhaps likely) that intruders claimed their land and then, in collusion with the corrupt Oklahoma courts, were given "legal" title at Statehood. (See And Still the Waters Run by Angie Debo).
The riverboat J.R. Williams, captured by Confederates in 1864, was carrying supplies to Ft Gibson and on its return trip was to stop at Webbers Falls to pick up lime from the lime kiln, across the river from the Webbers Falls community, and also to stop at the mouth of the Illinois River to pick up salt from Mackey's Salt Works, where the 2nd IHG was posted. Of course, the boat never made it to Ft Gibson.