Perhaps, the most significant (and first?!) example of such automatically generated combinatorial conjectures was Graffiti created by Siemion Fajtlowicz in 1982/3 or so. Later, he was joined by his student Ermelinda DeLaViña;
see https://en.wikipedia.org/wiki/Graffiti_(program).
Paul Erdos and other mathematicians enjoyed Graffiti.