Kenku can only speak using their mimicry trait but that means you have to have heard someone speak a phrase or word in order to speak it. Do I need to write down and keep track of all the phrases that I've heard so that I can communicate? Or can I only remember and mimic phrases I've heard recently?
In general, kenku are capable of a full range of regular speech. It's simply that that speech may be a mix of voices and sounds the kenku has heard (anytime) before. Essentially, they're kind of like a GPS's text-to-speech or another voice software that works by combining existing sound bites together. They can't create new sentences themselves, only piece together parts of what they've heard.
friend will be playing a kenku in a one-shot. We discussed how mimicry works. Are they more like a parrot, repeating what they've heard? or can they observe a group for a time and make sentences using said voice? how long can they remember sounds?
More about kenku: they can cleverly piece together voices and sounds they've heard to communicate. They're not less intelligent than other humanoids, but they do have an unusual form of communication. To simplify roleplaying one, think of two-three voices for the character.
So no, you don't need to track every single word you've heard before; in general, by the time you become an adventurer it's probably safe to assume you've heard the gamut of words used in normal conversations (and even some uncommon ones) - it's just that when you say those words, each word or phrase might be in a different voice (that of someone you've heard say that word/phrase before). And if you haven't heard a word/phrase before, you'd need to hear someone else say it in order to say it yourself... Or I suppose you could pronounce it syllable by syllable.
It might be fun to keep track of snappy one-liners or funny things so you can echo them in the other character's voice later.
iirc RAW states than kenku are capable of communication. that being said they are only capable of mimicking sounds. I would assume RAI means that even though they can only copy sounds for communication they still understand exactly what those sounds convey for the intent of communication.
personally I have interpreted that to mean that kenku speech is very similar to that of Bumblebee from the transformers movies: he uses all the correct words in the language he's speaking, but they're like sound clips from different voices.
so to answer your question, no I don't think its necessary to log all the phrases you know, though it would help with the flavor and rp of the character to know what phrases are spoken in what dialect/tone/etc if that's what you want to portray.