It is a mouth-full to say as a full name.
I found this description here.
hFE is an abbreviation, and it stands for "Hybrid parameter forward current gain, common emitter", and is a measure of the DC gain of a
junction transistor. So on a multimeter, it indicates a mode where
the meter can measure (probably crudely), the HFE of a transistor.
EDIT: When I talk about transistor gain with other engineer's, we often use the term 'beta', yet in a datasheet 'hFE' is normally what is used by the manufacture based on calibrated equipment for a standalone transistor.
For some transistors hFE readings may be done at several crucial frequencies as well as DC. 'Beta' is a better term for common-base designs, or just a general statement about DC and/or AC current gain in a known circuit.
As a refinement of the original answer, @carloc mentioned that a 'hFE' spelling refers to a DC signal of relatively large amplitude, while 'hfe' refers to a small signal measured deferentially around some common bias point. No specific thresholds were given, though my original answer refers to 'hFE', the DC gain of the transistor.