In the rule book, it said that title can add 3 to your side's total str in any power challenge.

So there are two scenes:

Does a character have power icon but have 0 str can be declared as a defender if the defend side have crown regent title.

A character have power icon be declared as defender(he is the only defender) in a power challenge, then his str has been reduced to 0, at that time dose crown regent's effect(+3) still work?

Yes. To both.

A character with 0 STR isn't prevented from participating in a challenge.

Note that the 0-STR character can be declared as a defender in the challenge even if Crown Regent is not your title. There is, as radiskull says, no requirement that your character have a STR greater than 0 in order to participate in the challenge.