Example:
They built him a [...] to commemorate his achievements and contributions to society.
Example:
They built him a [...] to commemorate his achievements and contributions to society.
Monuments and memorials usually do not depict the person, unlike a statue.
I would write:
They built a statue of him to commemorate his achievements and contributions to society.
or:
They built a memorial to him to commemorate his achievements and contributions to society.
If they're dead and it's at the site of their burial, it may also be called a tomb effigy or a gisant.