The circumpunct (circled-dot; dot within a circle) was indeed an important symbol for the Illuminati – in particular, it was used in internal correspondences, instead of writing the words “Order of the Illuminati”; similarly, when referring to a lodge a square symbol was used instead.
The symbol has a long history within Freemasonry, hermeticism, rosicrucianism, alchemy and astrology. It’s the alchemical symbol for gold but more anciently has always represented the sun.
For Weishaupt, however, his use of the symbol most certainly derived from the Monad symbol of the Pythagoreans, which was elaborated upon by Gottfried Leibniz in his Monadology of 1714. Weishaupt was greatly influenced by Leibniz and also revered Pythagoras and the ancient schools of wisdom. Look to what it meant to both Pythagoras and Leibniz and you can be sure that it was in this context that the Illuminati had used it.
– Terry Melanson (in response to a comment here)