It's important to note that:
* Ancient Greeks often had only one name. This was common practice, especially for men.
* Patronyms (father's name) or other descriptive elements were sometimes added. For example, we might find references to "Solon, son of Execestides" if needed for clarity.
So, while we don't know a "Solon" with a longer name, the name Solon itself was complete and sufficient for his identification in his time.