When I get a message from Twitter saying that one of my contacts is now on the service, I get their Twitter handle but no other information. Very often people won't use their actual name or photo, and I can't figure out who it is. How can I find out which of my contacts is being referred to?
The information available to you through Twitter will be only what that person has allowed to be shown on their profile, and it won't include their phone or email address (which Twitter likely used from your phone contacts to connect you two).
You could run a web search on their Twitter handle and see what related sites come up -- folks will often reuse names across different services and correlating those can provide you with enough context to figure it out.
... or, you could DM the user and ask who they are.