The short answer to whether she has a case is “no.”
Ms. Pierce is suing for invasion of privacy, defamation, and emotional distress. Let’s take these in reverse. She basically can’t win on emotional distress unless she wins on one of the other charges. This is because a charge of emotional distress requires one of two things: (1) the intentional infliction of emotional harm, or (2) the infliction of emotional harm through grossly negligent behavior. There’s no way for Pierce to prove (1), and in fact there’s far too much evidence against it (Ellen’s show doesn’t focus on her, and it has engaged in this sort of joke time and time again without incident). So this leaves Pierce with option (2), which for all intents and purposes is going to require that Ellen broke some other law that subsequently caused the emotional distress. A further complicating factor is that emotional distress requires physical harm. That is, Pierce’s distress must be so bad that it has caused prolonged symptoms. She must also prove the link between her emotional distress and those physical symptoms.
As for defamation, the joke is her name. But a defamation charge requires that the information shared be either false or (in some jurisdictions) grossly misleading. As “Titi Pierce” really is her name, it’s hard to say that the joke plays on anything false or misleading. Furthermore, there are protections in place for comedy and satire that make defamation charges exceptionally difficult to win. So even if Pierce’s claim is based on Ellen’s comedic mispronunciation, she is unlikely to get very far with that.
The only place where she has anything close to a case in the invasion of privacy charge. By showing the entire sign, including Pierce’s phone number, Ellen has arguably thrust her into an unwanted limelight. That said, Ellen has not intruded on Pierce’s private affairs, she has not publicly disclosed private information (Pierce’s first name is public, even if not widely known prior to this), and she has not place Pierce in a false light to the public (again, “Titi Pierce” really is her name). These are three of the four qualifications for invasion of privacy, and Pierce’s claim falls on each of them. The only possible case she could make is that Ellen has appropriated her name for commercial advantage. But this won’t work because (a) Pierce has already placed herself and the sign in the public eye (which is an absolute defense against invasion of privacy charges so long as no further crime has been committed, such as breaking into someone’s house to get information), and (b) those pesky laws protecting comedy and satire—the same ones that let Leno read every typo ever printed in the Palookaville Post—will again protect Ellen and her show.
@zenvelo “Public domain” refers to intellectual property. Pierce’s sign is not in the public domain. It is, however, in the public eye. The distinction may seem minor, but the law is built on minor distinctions.