About 5 years ago, my battery died in the underground parking of my office. I had a friend pull up his car to give it a boost, but nothing. Same as you described... just a bit of clicking. I had to call to get it towed to a local shop.
Turns out that the battery had shorted out, and I just needed a new battery. Not sure why it wouldn't start when being jumped from another vehicle, but it just clicked as you mentioned. I'm not sure if this is the same scenario for you, but maybe trying a new battery might help.
Maybe a Walmart or Costco one that you could return if this fails...
Just a thought, sorry to hear about your troubles.
Good luck.
Gearlist:
Electric: Ibanez 'AS103', Fender Dlx Nash Pwr Tele, Fender Squier '62 JV Strat, Squier '51, Squier 60's Classic Vibe Strat, Epi Elite LP Studio, Hagstrom Swede Acoustic: Larrivee LV-03RE, A&L AMI, Yamaha FG340-T Bass: Yamaha BB 450 Amps: Roland JC-120, JC-50, Peavey Classic 30, Fender Super Champ XD Pedals: Marshall Guv'nor Plus, Danelectro Cool Cat Drive, Transparent Overdrive, Digitech Digiverb, Bad Monkey, Ibanez TS-9, Boss AC-2, CE-5, CS-2, DD-3, DF-2, DS-1, FV-100, GE-7, OC-2, PSM-5, SD-1, TU-2, DVM~BYOC 'Lush Puppy' Chorus