Why is my iPhone 4 flickering after I replaced a home button?

Sounds like you may have damaged something on your way in when you replaced the home button.

I would start again, using this Guide to disassemble the phone and then ensure that all the cables and connectors were in tact and undamaged.

Then I'd clean the phone as instructed to ensure no oils from your hands remained.

Then I'd put it back together very carefully and hope it worked.

If the problem continues to persist or you find that you've damaged something, you'll have to replace it before the phone will work properly again.

Hopefully the damage isn't too severe and it remains salvageable.