Lucas Cost of Business Cycles I am trying to understand the logic behind Lucas attempting to analyse the potential benefits of eliminating business cycles by attempting to calculate the percentage of income that individuals would be willing to pay to eliminate such cycles/fluctuations;
My understanding is that Lucas proposed measuring the cost of business cycles as a percentage of consumption that would make a consumer indifferent between a world with and without business cycles - and that, this was found to be an unrealistically small proportion, from which it may be concluded that business cycle fluctuations are not very harmful for welfare(?), economic growth(?) etc.

However, how exactly is the impact of business cycles quantified in such a framework? In other words, if business cycles are so severe that many people lose their jobs, then if they were aware of this possibility, they'd pay a far greater insurance not to lose their jobs, i.e. to live in a world with no business cycles.
The Lucas model does not address at all the points that you address of people losing jobs, as it utilizes just one agent that simply has a perfectly steady (perhaps growing at a constant rate) consumption path. A model with heterogeneous agents where they have the possibility of losing their jobs and having some periods with, supposedly, very low consumption, would give a much higher cost of business cycles.

Further, in a paper using robust preferences Sargent and some coauthors find that if the agent has fear that his model of where income fluctuations are drawn from is misspecified, the costs of business cycles are much higher.