The monitor being ill,we'd better put the meeting off.为什么都用b

因为这句话是个简单句,简单句里只能有一个动词充当谓语动词.
那么,我们可以看出,这句话已经有谓语动词 had better put了,因此要用非谓语being.
如果你想说the monitor is ill,那么这句话就应该加入一个连词,如下：
The monitor is ill,so we'd better put the meeting off.（因为班长生病了,因此我们最好取消会议.）
The meeting is over,so/and all of us went home.（会议结束了,然后/所以大家都回家了.）