There are lots of pages that describe using Apple's Assistive Touch to work around the iPhone's defective-by-design home button . Alas, none of the articles I read told me how to get to the App Switcher (multitasking screen). On my son's balky iPhone 4 I can get a single-click to work, but a double-click is hopeless. Changing the Home-click speed didn't work.
Fortunately it's pretty simple. On iOS 6.1.3, after you've enabled the assistive touch "hockey puck" and moved it to a good location on your screen, tap once to bring up the main screen with the Home button, favorites, etc. Now tap on Device then again on "More". Multitasking at bottom will bring up App Switcher (really, it should have been labeled App Switcher - bad Apple). Unfortunately you can't create a custom gesture for App Switcher on a non-jailbroken iPhone; the iPad four finger sideways swipe doesn't work.
You can also have the home button repaired, but iFixit rates repair as "difficult". That translates as "elvish complexity" - normal humans won't be able to do this on their first attempt. Apple may replace an iPhone 4 for about $150, but Apple has been increasing replacement charges. FirstTech, a reputable independent repair shop in Minneapolis, charges $159 for a swap, but has no separate charge for home button repair. Note a device swap should include a very useful battery refresh - but do confirm that.
CNet's four ways to fix an unresponsive iPhone home button lists a connector bend and alcohol fix approach. Gentle connector bend had a minimal effect on my son's i4. I may try the alcohol fix. It's not worth paying $150 for a device swap as he's due to inherit a 4S when Emily goes to a 5S. His 4 will become a standby device.
 Apparently a flex cable problem, aggravated by bending of the power connector beneath the flex cable. Changes in the design of the iPhone 5 should make this much less likely.
- Four ways to fix an unresponsive iPhone home button | How To - CNET: Buried by link farm posts, this is the best summary of approaches: calibrate/restore (unlikely), realign (possible), clean (worth it) and assistive tech home screen.