I would stick with the truth. The truth doesn’t mean the whole truth, but some subset. At minimum you will feel better if you don’t lie, but worst case you get the job and your manager later learns there was no relative. You’ve abused trust and can harm the relationship.

You are allowed to be ill. You don’t have to elaborate on the condition. Just emphasize it was serious enough you were unable to work. Assure them you’re recovered and hence now looking.

But I would avoid the impulse to lie. It doesn’t matter if everyone else is lying around you. Or if corporations lie. Or whatever. There’s literally nothing to be ashamed of or to hide, so why descend to a level beneath yourself to hide something that doesn’t need hiding?

Finally there seems to be a view here that all managers are out to get you or extort you and will lie and cheat to get it done. I’m a hiring manager of hundreds and I genuinely do care about my people as people. When someone is sick I appreciate what that means. You shouldn’t want to work for someone who doesn’t. If the only way to get their job is to lie about something you shouldn’t have to hide, why on earth would you want their job? There are better people out there hiring. Even if it takes a little longer find the RIGHT job not the expedient job.