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À ma mère et que ce petit Patachou que je lui amène en le tenant par la main lui redise avec gratitude tout le bonheur de mon enfance

Redise is a conjugated form of redire (repeat, reiterate, tell again), here at the third person of the subjunctive present because it is a wish.

Here is one way to translate that dedication:

To my mother; and may this small Patachou, whom I bring to her holding him by the hand, reiterate with gratitude all the happiness of my boyhood.

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.

Addendum to the previous answers

Prefix “re” is very common in French (for a verb) and it indicates that the action was repeated. Some verbs with “re” have entered the dictionary but you can also create your own verbs, even with multiple “re”, and everyone should understand.

The common prefix is “re” (pronounced “reuh”) but it you add it before a verb beginning with the sound “é” (or “è”), “re” is pronounced “ré”. Example (note that I am a chemist):

J’ai extrait une première fois avec XXX mais ça n’a pas marché. J’ai tout ré-extrait avec YYY ; encore un échec. Du coup, j’ai ré-ré-extrait avec du ZZZ et ça a finalement marché.

Redise is the subjunctive of redire. And the tense is past subjunctive, not past perfect.

So the sentence would* translate:

“To my mother and to small Patachou whom I took by the hand and would re-tell much about my happy boyhood with pleasure.”

*Note the use of the subjunctive, would, here, as well.