I think this is an odd question, and I'm not certain what they are looking for. But I think they want
He has been ill since Monday.
Your suggestions don't work because we don't usually use the perfect with a specific time reference ("on" or "at" as opposed to "since").
Also, we don't often use the perfect with "become", so when we do it usually has some special connotation, often that the change is significant. So if we say that somebody "has become ill", we usually don't mean that they have just caught a cold, but probably that they have a serious or long-term illness.