He never tells the office guy what his name is in the current ad running…
It is difficult to find a definitive answer concerning whether the GEICO gecko has a first (or any) name.
GEICO’s page about the gecko’s origin does not discuss whether he has a name.
According to PetPlace, the Martin Agency, which created the gecko, says that he has no name:
Speaking of, what is that gecko’s name, anyway? I guess we’re going to have to wait to find out. The Martin Agency feels that the Geico gecko should keep a little mystery about him. So for now, he remains nameless.
On the other hand, AB Newswire claims that his name is “Martin”, taken from the name of the agency that created him:
A concept to use a gecko in commercials was born at GEICO’s ad agency, the Martin Agency, and he was named Martin after the agency. Martin first appeared in 1999-2000 television series during the Screen Actors Guild strike that prevented the use of live actors. In 2005 Martin was voted America’s favorite advertising icon.
I don’t know whether PetPlace or AB Newswire, or neither of them, is correct….