Replacing a broken home button with a virtual one can be achieved using stock iOS Accessibility settings, or with a variety of jailbreak options. In some cases, using a virtual home button is preferable to pressing the physical button even if there's no malfunction involved. vHome is a jailbreak tweak that provides a slick option to add virtual home and sleep buttons to iOS.
Once installed, vHome adds a representation of the iPhone 5 home button as an overlay to the iOS display. The virtual button will act just like the physical version, with a single or double tap duplicating the function of the real thing. vHome gets interesting when this virtual button is relocated to the top half of the screen. In this case, the button will change into a likeness of the sleep/power button.
Just like the real sleep button, tapping will lock the iOS display, and a tap and hold will initiate the shutdown process. Besides a toggle to completely disable the tweak, settings include a slider that will adjust the opacity of the virtual buttons. Changing the opacity simply affects how the virtual buttons will appear on the screen.
vHome is available on Cydia from the BigBoss repo for free. Jailbreaking iOS 7.1.x is now possible with Pangu. Check here for instructions on how to jailbreak iOS 7.0.x with evasi0n 7. You can also find out how to purchase and install Cydia tweaks with this guide.