They have the Tower of Hanoi puzzle in Rise of the Planet of the Apes but for some reason they call it the Lucas Tower.
Is there any reason why they would change the name of such an iconic puzzle?
The names are interchangeable :) The Tower of Hanoi aka Tower of Brahma or Lucas' Tower was "invented" (in the west) by the french mathematician Edouard Lucas in 1883.
From Wikipedia:
The Tower of Hanoi is frequently used in psychological research on problem solving. There also exists a variant of this task called Tower of London for neuropsychological diagnosis and treatment of executive functions. (...) The Tower of Hanoi is also used as a test by neuropsychologists trying to evaluate frontal lobe deficits.
However, I agree with you: in psychological research we rarely refers to it as "Lucas Tower". It's much more frequent to call it Hanoi Task or Tower of Hanoi.
The Tower of Hanoi was first popularized in the western world by a French mathematician named Lucas, so perhaps scientists are more likely to call it by its more formal name.