The two most memorable quotes from To Kill a Mockingbird regarding the bird are:

Remember it's a sin to kill a mockingbird.

and

Mockingbirds don't do one thing but make music for us to enjoy...but sing their hearts out for us. That's why its a sin to kill a mockingbird.

Both of these quotes state the same thing- that the things which bring us happiness and tranquility are important in life.

If one were to consider Boo Radley in the same way as the mockingbird, they could substantiate that he, too, tries to insure happiness (by giving gifts to the children) and tranquility (by protecting them from Bob Ewell).

The relevance of both quotes lies in the fact that the importance of the mockingbird is not known until both Atticus and Miss Maudie tell the children about it. Similarly, the children do not know the importance of Boo either, not until he becomes relevant to the children.

In To Kill a Mockingbird, why is Boo Radley a mockingbird figure?

The mockingbird is one that sings beautifully, but never hurts anything; hence the saying that it is a sin to kill one.

Boo Radley is mockingbird-like in that he never hurts anyone, but only tries to be helpful.  He had some issues in his past, but most of those can be traced to his parents and his older brother.  He is reclusive because he has been forced to be that way, and over the years it has become his way of life.  But it isn't because he doesn't love other people, or because he doesn't want to be a part of the world.

Examples of Boo's kindness abound in the book, in the ways he treats the children.  He leaves treats for them in the knothole of the tree, he puts a blanket around Scout's shoulders during the fire, and eventually he saves them from death at the hands of Bob Ewell.  He loves "his children," and would do anything for them.

In the end of the novel, Atticus and Heck Tate discuss Boo and what to do about his involvement in the Ewell case.  They decide to keep it a secret, because to expose Boo would be to drag him into the spotlight, where he would be very uncomfortable.  Like killing a mockingbird, it would be a sin to do anything to hurt Boo Radley, after all the good he has done.

In To Kill a Mockingbird, why is Boo Radley a mockingbird figure?

The reason that Boo can be seen as a Mockingbird figure all goes back to Atticus' warning that it's a sin to kill a mockingbird.  This was when he gave Jem his first BB gun, and was giving him rules about what he could and couldn't shoot.  He says killing a mockingbird is a great sin, because all they do is make beautiful music all day.  In other words, it is an innocent, harmless creature.

Boo can also be seen as a harmless, innocent creature as well.  He has never harmed the children, and in fact, even saves their lives in the end when Bob Ewell attacks them.  He almost adopted them as his own, giving them surprises in the tree knot hole as his only social interraction with anyone in the outside world.  He preferred a life of seclusion, as this was all he was used to.  After Boo killed Bob Ewell, Sherrif Tate refuses to publicize the heroic deed, because it would draw unwanted attention to Boo, and cause him added stress and woe.