What could a candidate do/say during an interview to prove his stability in the job term offered to him?

Its very good question.

Answer to this question can be given only if you have done good research on co, role, culture.

You have to relate with your strengths, aspirations and It should be facts,

Like what are the few things by which1 employee stays longer in an organization

then you can say that i spoke to few resources of your co, read reviews and ppl say.

rest i got to know about it by u  etc etc.

So create a justification based on natural facts.

I will stay in your co that , this is kind of role and co I was looking for and I see there is a very strong match.

He/sheshould say the he/she is passionate about working in their organization, withtheir knowledge and skills they can help achieve the organization aims objectives and goals while building their career and show that they will become very effective asset for the Company