Tyger Tyger... isn't just a code. The first time we hear it is when Red John himself recites the first line of Blake's "The Tyger" to him while saving his life. But why does RJ share this ambiguous info with Jane? Two reasons.

1) He's saying, "I've got your back." The Tyger Tyger ring of law enforcement officers have instructions not to kill Patrick Jane. Maybe RJ thinks he can win him over, and he would relish being able to do that; or else, He really thinks his web is so well woven that Patrick will never catch him and eventually give up, and he relishes in that too. After all, RJ does reach out to Jane, in "Crimson Hat" after Jane pretends to have given up. 

2) He knows that if Jane stays on the trail long enough, Jane will eventually come to know of Tyger Tyger. Once Jane comes to know about the code word he will begin to realize how vast RJ's web of influence is, and be daunted by this discovery.

Finally, Red John makes mistakes... Jane will use this information to ensnare him in his own web.