Why did the time traveller write his name on the statue he found in the museum?
The Time Traveller does this in chapter 8, when he explores the Palace of Green Porcelain properly for the first time and realises that it is an old ruined museum. This ties in with the overall theme of the book. Museums look back in time while the Time Traveller has gone further forward in time then anyone could have imagined. Being in the musuem gives him an even longer perspective on time, stretching way back to antiquity. The result is that he is overcome with a kind of exhilaration, and so scribbles his name on the statue with a kind of childlike glee. He also wants to leave his own visible mark in some way in this future world, although no-one else is ever likely to acknowledge it.