The best alternatives to Birthday Wiki are Birthday App, TweetStory, and #TwitterThrowback. If these 3 options don't work for you, we've listed a few more alternatives below.
Take your friend’s birthdays and LEAVE FACEBOOK FOREVER. Birthday App finds all your contacts and friends' birthdays and create a calendar. You don't have to lift a finger.
It's like Facebook's "On This Day", but for Twitter. The easiest way to look back on your Tweets from the past. Source code: https://github.com/shalvah/oldtweets.today
Timeflower is a simple tool for your photo library. It supports iPhone and iPad. • On this day: View your photos taken on this day throughout the years. Today's news is yesterday once more. • Favorites: The best photos, curated by one and only you.