If I stick the right size and length wire into the center conductor of the male BNC on the HT I instantly have an antenna?

No other connections to be made somewhere?

A more general question; what happens if the wire in the center conductor bends over and touches the barrel on that BNC?

Yes, you have an antenna. 19 1/4" is the length IIRC for a 2M.

You don't have a counterpoise or ground plane, but neither do the OEM antennas. Generally your body is a capacitively coupled counterpoise.
Some people actually add a wire to the barrel side of the connector as a counterpoise and find it improves thisn greatly.

What happens if it shorts out will depend on the radio's design. Anything from nothing at all to a blown final transistor.

FPNI

ok great.  lots of google results for the tiger tail but few details about the basic design of an HT antenna.  Too basic I guess .

So I could solder some RG-58 center conductor onto a tape measure blade, crimp a BNC center connector onto the other side of the conductor and slide that into the BNC center.

Yep.

Inside a rubber duckie.

Connecting tape measure to the center BNC conductor + some epoxy or silicon and you're good to go with no worries of shorting out.

Also keep in mind they make really small tape measures with metal tapes, and that self-sealing rubber electrical tape sticks to itself if you wanted to protect the tape measure.

Yes, any conductor is an antenna.

A wire of the appropriate size to make good contact with the center conductor of a BNC socket will function as a 1/4 wave antenna just as any other conductor plugged into that connection. Something sold as an "antenna" is just mechanically more rugged.

If you want to experiment with connecting wires and such to a BNC connector I'd suggest a BNC to Banana plug test adapter:

Attached File

In general, the optimum length of wire is going to be significantly longer than 1/4 wavelength.  The system ends up being an OCF dipole-ish structure with typically your hand acting as an inadequate counterpoise.

Antennas for hand-helds are (or at least should be) designed with this in mind, which is why the "tiger tails" as often do nothing or even degrade performance by detuning, as often as they may help.

Some of the crappy chinese replacement antennas I once checked out apparently didn't understand this issue, and were tuned to the HT's operating frequency (some were even higher up into the business bands above the ham bands) - when they should have been tuned somewhat lower.  That was why they had poorer link performance than even the shorter (but correctly tuned) stock unit that came with the radio.

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What is the diameter of the center conductor (that itty-bitty pin) in a 50 ohm BNC connector?

Your choices are:

.024

.031

.039

If none of the above then mouser.com doesn't have it in stock and could you please tell me the right diameter and where to get some

Since I switched HTs I don't have many BNC ducks, but I just got the calipers out, and the
one on my scanner actually measures .042, and the right angle adapters I use for my VHF
rig are .052. So I'd go with the biggest one (.039) if that's any indication.

Just get a BNC connector with a non-captivated pin and you'll have the right size.