`Foo factor' and `foo quotient' tend to describe something for which the issue is one of presence or absence. The canonical example is fudge factor. It's not important how much you're fudging; the term simply acknowledges that some fudging is needed. You might talk of liking a movie for its silliness factor. Quotient tends to imply that the property is a ratio of two opposing factors: "I would have won except for my luck quotient." This could also be "I would have won except for the luck factor", but using *quotient* emphasizes that it was bad luck overpowering good luck (or someone else's good luck overpowering your own).
`Foo index' and `coefficient of foo' both tend to imply that foo is, if not strictly measurable, at least something that can be larger or smaller. Thus, you might refer to a paper or person as having a `high bogosity index', whereas you would be less likely to speak of a `high bogosity factor'. `Foo index' suggests that foo is a condensation of many quantities, as in the mundane cost-of-living index; `coefficient of foo' suggests that foo is a fundamental quantity, as in a coefficient of friction. The choice between these terms is often one of personal preference; e.g., some people might feel that bogosity is a fundamental attribute and thus say `coefficient of bogosity', whereas others might feel it is a combination of factors and thus say `bogosity index'.