Chosen Solution
My home button does not want to work lately. I recently replaced my defective battery with a new battery that I bought from ifixit which solved my battery issue. I replaced the battery about a month ago and now all of a sudden my home button does not want to work. It may just be a coincidence but I do not know enough about this stuff to say one way or another. I just want to know how to fix this issue and if it is even worth while. The phone is roughy 4-5 years old.
Its possible that when you replaced the battery, the home button got misalligned. Pop open the screen and give it a look. Otherwise, a new home button on an iPhone 5 is one of the easiest repairs ever. No Touch ID to mess with, just pop the old one off and throw on a new one.
This is more likely to be board damage. The ifixit guide of prying the battery is stupid and usually damages logic boards.