In the right hands, they certainly can. Chevrolet spent years advertising its cars using that “heartbeat of America” jingle, a tune that's remembered fondly by an entire generation. On the other hand, General Motors is one of the biggest companies in the world, and it could afford to scour the globe for the exact right piece of music to use in its advertising campaign. The chances are that your law firm's pockets aren't quite as deep!
If you've ever watched daytime TV, you already know that most jingles are unmitigated disasters—usually a seven-digit phone number or the company's name sung in near-random notes by a poorly rehearsed chorus. Some of these jingles are so bad that they're also, in their perverse way, very effective, but most of them are so bad that they're...well, just bad. Simply put, they won't do anything for your firm's name recognition, and they might even prompt viewers to mute their sets (or, better yet, get up and go to the refrigerator) whenever your commercial comes on.
Rather than silly gimmicks like jingles or animated mascots, you're better off putting your best foot forward and telling the viewers of your TV commercial what you can do for them in their hour of need (“Have you been in a car accident? Call our firm to learn the ten mistakes that can hurt your personal injury lawsuit!”)
Ideally, your TV commercial should aim higher than simply making a potential client look twice: it should be so sincere, and so convincing, that that person remembers your name weeks or months down the road when he's in need of a lawyer.