I really only have two. First, don't choose a name that might draw attention, such as, let's say, "Euphonius Q. Supertramp," the name on a piece of spam email I got a while back. A name like that, while awesome, is probably going to lead to some sort of investigation. Second, if there is a warrant out for you, don't choose a name that belongs to someone else who is also wanted, because that totally defeats the purpose.
In Great Falls, Montana, Jonothan Gonsalez gave the name "Timothy Koop Jr." during a traffic stop last Monday, only to learn to his dismay that there was already a warrant out for "Timothy Koop Jr." Gonsalez was mistakenly arrested on the Koop warrant, but the mistake became irrelevant when the search incident to the mistaken arrest turned up some real meth.
That report didn't say why Gonsalez had chosen that particular name, but we know why Mario Miramontes decided to give his cousin's name during a traffic stop in Dallas -- he said he thought his cousin had a clean record. As it turned out, not only was his record unclean, he was wanted for allegedly "fondling an underage relative." ("D'oh!" is a Simpsons trademark or I would have a category devoted to it already.)
Miramontes said he had figured he would still be released once the error was discovered, but this also did not go as planned, evidently because he got arrested in Dallas where it is apparently possible to be held for 13 months without access to a lawyer or judge to explain a mixup of this kind. Miramontes is now suing the county for what he says was an illegal (or at least illegally long) detention.
Miramontes did write a letter to a judge at some point during his detention saying he had been arrested "because the police thought I was my cousin." It wasn't clear whether he mentioned that this was because he told the police he was his cousin, but the letter didn't work anyway.