Yeah. Hi all. I recommend you go with the spellbook as per wizard. A familiar is fine. That's what they do. It's funny though, it just highlights how bad the beast master ranger is by comparison, who requires either an action to command their animal companion or a bonus action at level 6. I love the montage by the way! Everyone else should wake up when they want to. I will reveal loot information when one of your characters decide to investigate that.
Okay, I'm going to make my pitch for the Arcane Trickster to not have a spellbook (mainly because it's 50gp and I'm a little strapped at the moment after buying the component pouch and components for the familiar). I'm good either way you rule, but the little part of me that has a list of stuff that I'm saving up to buy wants that 50gp.
[SBLOCK=The Pitch]The most direct reference I can find is
this twitter exchange with Jeremy Crawford. "As written, the Eldritch Knight and the Arcane Trickster don't use a spellbook. They also don't prepare spells." That's clear to me, but the case from the PHB is a little quieter.
In the PHB itself, the Eldritch Knight is specifically called out for not using a spellbook on pg.75. It is a 1/3 spell progression class just like the Arcane Trickster. In fact, the language describing the mechanics of their magic is identical with the exception of allowed schools of magic. The specific quote for Eldritch Knight is this, "These knights learn a comparatively small number of spells, committing them to memory instead of keeping them in a spellbook."
The Arcane Trickster does not feature that specific line, but it also never mentions a spellbook. The Arcane Trickster and Eldritch Knight seem to be mirrored options of one another. It seems like in 5E, they made the choice to make the spellbook a specific feature of the wizard to show its extreme flexibility. An Eldritch Knight or Arcane Trickster does not have any way of learning new spells other than gaining levels unlike a wizard with a spellbook who can potentially gain access to every spell on its list. If you're like me and have played prior editions, this seems counterintuitive because I assumed Vancian style casting (fire and forget). That doesn't exist in this edition (notice how a wizard can keep casting his prepared spells multiple times using his spell slots). With that in mind, the idea of a character memorizing a comparatively small number of spells without a book makes sense.
Okay, I'm going to take off my rules arguing hat. Hopefully, the above comes across positively. I'm genuinely good either way you rule. You're the DM. I just want to save some coin.[/SBLOCK]