Big Dumb Objects often exhibit extreme or unusual properties, or a total absence of expected properties:
The monolith in Stanley Kubrick's 2001 A Space Odyssey is an indecipherable influence on the protohumans to whom it first appears, and later in the film serves to show how little humans have evolved. Astronaut Bowman's attempt to interact with the monolith only make him a part of its mystery.
The object discovered in Quatermass and the Pit was made of a material of extreme hardness, such that diamond-tipped drills and acetylene torches would not damage it. At the same time nothing would adhere to it.
In the movie based on Michael Crichton's novel Sphere, the eponymous object would reflect everything in its presence except people. If it did reflect someone, she or he was alone, and the individual was accepted as worthy to harness the device's power.
The eponymous Dome from the Stephen Kingnovel and television show. The dome is large and transparent unless touched by a person; it gives a slight electric shock when touched for the first time by someone, but not afterwards. It cannot be penetrated, even by a MOAB, and is seemingly causing a lot of mysterious things in Chester's Mill, the town that the dome is enclosing, including causing all electronic devices near it to explode, visions, and, in one character, premature birth. Such unexpected properties are usually used to rule out conventional origins for the BDO and increase the sense of mystery, and even fear, for the characters interacting with it.
^Nicholls, Peter, 2000, Big Dumb Objects and Cosmic Enigmas: The Love Affair between Space Fiction and the Transcendental, in Westfahl, Gary (ed), Space and Beyond: The Frontier Theme in Science Fiction, Greenwood Press, p. 13. "... I decided to write an April Fool's entry. I would pretend that a phrase I’d always liked, originated by the critic Roz Kaveney but not in general use, was actually a known critical term. I would write an entry called 'Big Dumb Objects' in a poker-faced style, suggesting an even more absurd critical term to be used in its place, 'megalotropic sf.'"