My first name is [name]Karen[/name], so I like the idea of passing on my name. However [name]Karen[/name] is derived from [name]Catherine[/name], which has a nice ring to it. Which do you like best and why? Last name starts with a “K”.
I think [name]Lily[/name] [name]Catherine[/name] flows much better than [name]Lily[/name] [name]Karen[/name]. It’s a nice way to pass on your name, but with a bit of a twist so it’s more original.
Good luck with whichever you choose. [name]Lily[/name] is a lovely name you can’t go wrong with! 
[name]Lily[/name] [name]Catherine[/name] is beautiful. [name]Catherine[/name] over [name]Katherine[/name] only because your last name begins with K.
All three are very nice, so I will just go on my personal opinions. I like the idea of mother and daughter sharing names (fathers and sons do it all the time!) but I’m not sure I would want to give my daughter my exact name in the middle. What if she decides she hates her first name, and now she can’t even go by the middle because it’s her “mom’s name”? So, I favor the variations of [name]Catherine[/name] for that reason only (not very important though). Based on looks alone, I like [name]Lily[/name] [name]Catherine[/name] over [name]Lily[/name] [name]Katherine[/name] (which is odd because I favor [name]Katarina[/name] over [name]Caterina[/name]), but it’s nothing more than aesthetics. Good luck in choosing!