The word redise is a form of the verb redire (subjunctive present). Redire follows a systematic construction: the prefix re- meaning “to do something again”, plus the verb dire meaning “say, tell”.
The construction in this sentence is rather difficult... The key word is the first que, and it's drowned in the mass of words. Let's start simple:
Que Patachou fasse quelque chose.
The word que has many different types of usage. With most of them, que starts a complement of a preceding verb or noun. Here, que starts a standalone sentence. Que follows by a verb in the subjunctive mood expresses an order or a wish. Here, it's a wish: the speaker wishes that Patachou does something.
Now let's add the part that says what Patachou is supposed to do:
Que Patachou lui redise tout le bonheur de mon enfance.
i.e. literally may Patachou tell her again about all the happiness of the speaker's childhood. Here the verb dire is not about saying a specific thing, but about conveying an idea, a feeling. Think “tell the tale of” rather than “tell” in English — “May Patachou tell her again the tale of the happiness of my childhood.”
Now we add complements: add more qualifiers about Patachou, and add “avec gratitude” to qualify redire. Here's the sentence with key syntactic groups marked:
Que {ce petit Patachou {que je lui amène {en le tenant par la main}}} lui redise {avec gratitude} {tout le bonheur de mon enfance}.
Finally “À ma mère” ((dedicated) to my mother) and that sentence are joined together with et.