Electronic – Drive LEDs from USB

This may be overkill for what you want, but Microchip do an Accessory development starter kit for Android.

With that you can create anything you like to interface with your Android phone. You can even use Arduino shields with it!

To this you would need to add something to actually drive the electromagnet. You might like to check out the many tutorials online about interfacing motors and solenoids to the Arduino, as the same applies to this board.

Most commonly either a transistor, or transistor pair (known as a Darlington Pair) is used, depending on power requirements. These are also available in an IC form, such as the popular ULN2003 chip with 7 Darlington Pairs in it.

I have created an Arduino shield that contains an 8-channel Darlington IC, the ULN2803, which you can etch yourself using Toner Transfer. The details on it are here.

_{Shameless self promotion there :)}

There's pretty much no chance you'll get all this for $4, it is not a realistic budget for this. But that's not to say it's not possible. You do need a microcontroller with USB functionality (or have it bit-banged, but USB ones are cheap enough) to talk to the Android phone. Then the Android phone could use its USB API in the SDK to talk to it and issue a command. When the USB microcontroller recognizes the 'take picture' command or whatever, it could flip a transistor on/off to toggle those wires you have and tell the camera to take a picture.

But you mentioned an 'old' Android phone, that could be a problem because only newer ones running Android OS 3.0 and up have the ability to expose the USB API to talk to it. If your target phones don't run that, this isn't going to work.

In that case, to accomodate most Android phones, even older ones, you could do this over Bluetooth. But that's going to raise your costs even more because you'll need a bluetooth module and power source for the Bluetooth, probably a battery, on top of the micro. But Bluetooth modules can be very cheap, here's one for $5.50 USD and they're even less in larger quantities: http://imall.iteadstudio.com/prototyping/basic-module/im120723010.html

Otherwise an even simpler approach with no micro could be making a circuit with a photosensor such that it triggers the camera wires to cross when it detects some amount of light, you could vary that with a pot/resistor. Then tape/connect the photosensor somehow to your phone and program it to light up the LCD or the flash on the back (if it has one) when it wants to trigger the photo. That might be do-able for under $4 but it's not very elegant.

Good luck!