There are two claims to the roots of Teddy Bears

AMERICAN CLAIM

In 1902, in America, Theodore Roosevelt was the President. Inventor Morris Michtom, who was a Russian imigrant in New York noted President Theodore Roosevelt's refusal to shoot a cute little bear cub in Mississippi while bear hunting in 1902. A cartoon by Clifford Berryman depicted the scene, showing the president with a bears head in place of his own turning away from the bear cub. Michtoms wife made a soft jointed bear which she displayed in the window of her sweet shop. Michtom having sort permission to call it "Teddy" ( which is short for Theodore) So it became "Teddy Bear". Roosevelt then used a bear as his mascot.

GERMAN CLAIM

Mararete Steiff of Giengen South Germany is said to have created the first Teddy Bear in 1903. Margarete, who used a wheel chair due to polio as a child, created some elephant pin cushions as family presents. She went on to do different animal toys including the first Bear standing on all fours in 1884, which she then sold at local fairs. She decided to set up in business with her Nephew Richard, who devised the first jointed bear, which was Exhibited at the 1903 Leipzig Toy Fair. It attracted a large US order. Steiff claimed that when President Teddy Roosevelt was unable to name the bears breed they were dubbed "Teddy's Bears". They later were called "Teddy Bears"

I’ll leave you to decide which story you think which is the correct claim.