`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'.
