My wife's employer changed names but it's still the same business do I need to delete the old W-2 info and replace it with the new one?

Yes, you need to enter the new information. If they changed their name mid-year and your wife received two W-2's, make sure you enter both.

Click on Where do I add or enter a second W-2?

Yes, you should change the name. But most importantly is the EIN the same? This is the Employer Identification number. 

Wait for the W2 to come.  If you get 2 and the EIN number in box b is different then enter both of them separately.