The expedition's plan had always been to trade with the Indians for horses once they reached the headwater's of the Missouri. This was necessary to cross the mountains they knew they would encounter on the way to the Columbia River. (In case anyone is wondering, they knew that the Columbia existed somewhere across the Divide because the first 100 miles of the river from the West Coast had been recorded and charted by maritime explorers)
However, it had been literally months since they had encountered any Indians at all. It's a mystery as to why that is. Were Indians intentionally avoiding contact or was it just such a large tract of land with a relatively small amount of people?
In any case, Lewis & Clark were starting to get desperate. There was a ray of hope when they reached the headwaters because that is where Sacagawea had been kidnapped as a girl by raiding Hidatsa. They now knew that they were close to her people, the Shoshone.
The confluence of three small rivers (the Jefferson, the Madison and the Gallatin) form the Missouri. The explorers named those small rivers after the President, the Secretary of State and the Secretary of the Treasury accordingly.