Home button replacement cost for iPod touch 5
How much does it cost to have the home button replaced on an iPod touch 5?
iPod touch, iOS 7.0.4
If not underwarranty Apple will exchange your iPod for a refurbished one for $99 for 16 GB 5G and $149 for the other 5Gs. They do not fix yours.
A third-party place like the following maybe less. Google for more.
iPhone Repair, Service & Parts: iPod Touch, iPad, MacBook Pro Screens
Fix it yourself if you are up to it