Your name on your biometric card must match the name on your passport.
If you haven't updated your passport yet, wait and do that within 90 days of your ILR application. Then you don't have to apply for a new card (which can take 6 months) and instead just apply for ILR in your new name. The new biometric card will be issued in your new name.
If you have updated your passport, you'll have no option but to update the biometric card. To avoid the six month wait, you can pay an additional £590 for same day service.