What would cause...

a TT to play perfectly beautifully but, on occasion, emit a loud, constant "hum" through the left channel only? Trouble-shooting suggests that jiggling the tonearm cable lead coming out of the back bottom of the plinth can set off the hum and also disrupt the hum. This is a Rega model RP3 just purchased brand new. Should I insist upon a return or should I wait and see if the problem resolves?
I'm sure you've already checked the tonearm wire connections so I'd take
it back asap. You don't want to buy a problem. If you take things back
right away, they'll probably replace it with a new one. If not, you'll
probably have to wait for them to repair it. That's not what you want... to
wait, and even then end up with a repaired turntable.
Return it. That problem is a warranty issue, and I would skip trying to have someone 'fix' it and just send it baack for a refund.

Some wire in th arm is frayed or pinched. No good is going to come of fiddling with it.

If it was not new, then yes finding and fixing the issue would be worth doing.
Wait. Think of upgrade too. I had older model, but I believe the new one also have motor EMI isolation issue. Overall, I was not impressed of its performance.
Take Liz's advice
Everyone seems ready to jump all over the seller without giving things a good look. I would think the first thing to check would be the RCA jack at the pre-amp. Wiggle the plug and see if that makes a difference and also try coating the plug with De-Oxit. Problem may also be a loose pin in the cartridge. Try it in another tonearm. Trouble may well be in the tonearm wiring but I'd check the simple and obvious first.
Sounds like an intermittent electrical wire/connection issue inside the table. That would be a defect!

If new, I'd take it back to the dealer to assess and address the problem, including making sure cartridge setup including wire connections is good as well. There should be no intermittent noise issues that result from jiggling wires (within reason).

In lieu of physically taking the table back to the dealer to address, I would lean towards contacting the seller to return, and see what they can offer up in terms of customer service to your liking from there. COuld range from tech support coaching you through, to a flat out return or exchange, whatever is needed to make a happy customer.
All previous posters a/o Mapman 10/1/13 are correct. The arm has a ground fault in the left channel. Could be in the arm itself, could be the RCA cable, could be the headshell, could be the cartridge leads, could be all of those. Not worth fooling around with yourself.
Thanx to all of you for your erudite responses. Dealer will take care of me, it seems.