I don't think they got the idea from Alice, from what I gather the idea for the giant onstage Eddie came from old Fan Club runner Keith Wilfort, who saw some giants at a carnival and suggested the idea to Rod and the band in 1981.
The first onstage Eddie that wasn't just someone in a leather jacket and mask was on the Beast on the Road tour in 1982
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Plus, Eddie is way cooler than a cyclops
*slight bias because I'm a diehard Maiden fan
Edit: just checked, it wasn't Keith Wilfort, it was their old lighting engineer Dave 'Lights' Beasley who suggested it after seeing a pantomime version of Jack and the Beanstalk, point still remains though