You could edit out the birthdate information from those contacts you do not want to see in the Birthdays calendar.
Another way to accomplish this would be by manually adding the birthdays you do want to see to a new or existing calendar and then disable the Birthdays subscription.
This appears to be an issue when Google is set up as a Microsoft Exchange account type (Google Sync) rather than the normal Gmail account in the Settings app. You can check that this is the case for you by opening the Settings app, going to the Mail, Contacts, Calendars section, and tapping on your Google account. If you see Exchange at the top of the page - rather than Gmail - then this applies to you.
It seems that in fixing an Exchange bug Apple broke Google's Microsoft Exchange implementation.
It also seems only to be a problem with secondary accounts.
As a temporary measure – until either Apple or Google have fixed this problem – you could open the Settings app, go to the Mail, Contacts, Calendars page, and Add Account... of type Gmail. You only have to turn on Calendars for this account, as the rest of the Microsoft Exchange implementation seems to be working. Then go to your original Microsoft Exchange account and turn off Calendars. You will be warned that all your Exchange ActiveSync calendars will be removed from your phone; choose to delete them. Finally, you might want to adjust the Default Calendar in the Calendars section of the Mail, Contacts, Calendars settings page
A few caveats with using a Gmail type account for calendars:
- Google event invitations will no longer work
- Syncing for calendars is now pull, rather than push; i.e. changing an event on another device will not be reflected immediately in your iOS calendars
Also, beware that events you have declined will not appear in iOS, and you may have accidentally declined events while getting frustrated at the iOS 6.0.1 calendar problem. These you will need to fix in the Gmail Web Client.
Best Answer
Settings > Mail, contacts, calendars > Default alert times > birthdays
Apparently there are a few options here. You cannot specifically set a custom time at the time of my writing this, but it is possible to pick "a day before", "2 days before", "1 week before" or on day of event. I don't see a way to select a different time though other than 9 am.