My husband wants to adopt my son and my son's biological father is willing to give up his rights. Roughly how much will this cost me to have a lawyer do the paper work for him to give up his rights and for my husband to adopt? Also once my husband adopts my son will his last name automatically change?

Less than 1000 in Brevard County FL, but your mileage may differ where you live. Brevard has very high court fees and the majority of that cost is court related and not lawyer.

His name won't 'automatically' change but it can be handled at the same hearing.