Bug-eyed monster is an early convention of the science fiction genre. Extraterrestrials in science fiction of the 1930s were often described (or pictured on covers of pulp magazines) as grotesque creatures with huge, oversized or compound eyes and a lust for women, blood and/or general destruction. The term is now often abbreviated to BEM.
In the contactee/abductee mythology which grew up quickly beginning in 1952, the blond, blue-eyed and friendly Space Brothers of the 1950s were quickly replaced by small, unfriendly bug-eyed creatures, closely matching in many respects the pulp cover clichés of the 1930s which have remained the abductor norm since the 1960s.
Artists Boris Vallejo and Julie Bell are famous for their artwork in this genre.
While reality caught up with Pre Astr Science Fiction in 2089 when the Sarans landed on Earth and were human as can be, they did come to Earth to warn of impeding doom caused by the threat of a Xunx Invasion. Xunx very closely resembled Bug Eyed Monsters (and Bugs in general). United Earth literature and common terminology of that term was full of related terms describing the Xunx.
Even famous Richard Stahl called the Xunx BEMs in his early reports.