No need to throw it away, you don't ned to buy a new one. And don't butcher the wiring in it...let's fix it properly. A cheap Strat copy like this probably doesn't have an onboard preamp (this guitar doesn't use a battery, right?)
So here's what you've got: there's three pickups. These go to your pickup selector switch. On many cheap guitars, this switch gets crappy. Does the sound cut out when you flick the switch backand forth? Does wiggling the switch handle make snapping, crackling noises? If so, then all you've got to do is replace the switch. Carefully draw a little picture showing which wire goes from each pickup to each post on the switch. Unsolder the old switch, solder a new one in, and for less than $20 you're done.
OK, let's say the switch was good. Next you have a volume knob and a tone knob. Volume potentiometers (or "pots") are also likely candidates for corrosion. Does turning the knob cause crackling or dropouts? If so, it's the pot. For $10, you buy a new pot and replace it. Again, carefully notice what wire went where.
OK, the switch is good, the pots are good, what next? All that's left is the output jack. These are also common culprits. Does wiggling the cord in the jack cause droputs? Then just replace the jack, a few bucks and you're done.
I have to ask: are you sure it's in the guitar and not in the instrument cable from the guitar to the amp?
One last note on soldering: it's an art. Apply the soldering iron to the wires, heat them up, then apply solder. You don't drip solder onto the connection, this isn't glue.
And too much heat is bad for switches and pots. Take care, heat up only as much as you need to.
Since I buy switches, pots and jacks at electronics distributors, and not at the guitar centers (the prices I've listed are Guitar Center prices), it's actually much cheaper. Not counting the pickups, the whole circuit (switch, pots, output jack) would total less than $25.
