According to General Sherman's memoirs, his father had "caught a fancy for the great chief of the Shawnees,". Therefore, Sherman writes, "When, in 1816, my brother James was born, he insisted on engrafting the Indian name "Tecumseh" on the usual family list. My mother had already named her first son after her own brother Charles, and insisted on the second son taking the name of her other brother James, and when I came along, on the 8th of February, 1820, mother having no more brothers, my father succeed in his original purpose, and named me William Tecumseh."

Sherman's father died in 1829, when he was still young, so there was not much time for him to ask his father why he was named Tecumseh, and soon after that the family scattered. Ten kids was to much for his mother to handle, and Sherman went to live with a Thomas Ewing, before going to West Point.

Charles R. Sherman, William Tecumseh Sherman's father, was a lawyer in Ohio and was appointed a Judge in its supreme court before his death. Most likely he was only a commissary in the War of 1812, when the United States fought the Shawnee. That means he would have been in charge of buying supplies and getting them to the troops. General Sherman's memoirs don't have a lot of information on his father, so there is very little information on why Charles Sherman was so impressed by the Indian chief.

Source

Memoirs of General William T. Sherman by General William T. Sherman- e-book ISBN 13 978-1-4209-5080-9 Second edition. The quotes come from pages 9 and 10.