A yo-yo was a round toy which was spun up and down on a string.
When the Second Doctor visited the Panjistri homeworld, he had a yo-yo in his pocket, something which was also present on his visit to Gandor. (PROSE: Timewyrm: Apocalypse, AUDIO: Lords of the Red Planet)
The Third Doctor, realising that the Time Lords had programmed his TARDIS to always return to Earth, compared himself to "some kind of a galactic yo-yo". (TV: The Claws of Axos) On Silurian Earth, a yo-yo was found in the pockets of the cloak on the Third Doctor's body. (PROSE: Blood Heat)
The Doctor's fourth persona was the one who most frequently kept a yo-yo on his person. Although he once used it to test local gravity, (TV: The Ark in Space) he apparently used it primarily for fun. His skill with the yo-yo appeared to increase over time. (TV: The Android Invasion, The Brain of Morbius)
At the end of his life, the Seventh Doctor had a gold yo-yo on his person. Chang Lee stole the yo-yo, along with the rest of the Doctor's possessions, from Walker General Hospital. He later returned the property to the newly-regenerated Eighth Doctor. (TV: Doctor Who)
The Twelfth Doctor later tried to use a yo-yo to impress a Viking village into believing he was Odin, quoting Clarke's Law. According to Clara Oswald, this was something that the Doctor did a lot to get them out of tricky situations. (TV: The Girl Who Died)