Is it me or I?

Get rid of Lucy in the sentence; that's how you figure this out. It depends on I to be honest. Does that sound right? No, you'd say it depends on me to be honest.

It's 'Lucy and me'. If you're not sure which pronoun to use, split the sentence into two sentences, like this:
1. It's important for Lucy to be honest.
2. It's important for me to be honest.
So, you can (I hope) see that the full sentence should read: 'It's important for Lucy and me to be honest.' 'Lucy and I' just wouldn't look right, would it?

You're confused because the verb "to be" normally uses "I" instead of "me." In this case, however, you're dealing with a prepositional phrase, so use the object "me."

Try eliminating "Lucy" from the sentence and listen to how it sounds.

"It's important for I to be honest." Uck.
"It's important for me to be honest."

Lucy and me. You wouldn't be able to say "It's important for I to be honest", so the use of "I" is clearly incorrect.

I am always astounded at the people on here who answer questions when they have no idea what they're talking about! Bookbinder and DWRead are right- it's "me"- object of the preposition.

for me
so,
It's important for Lucy and me to be honest.

It's important for Lucky and I, to be honest, is the correct answers.

ME! It's the object of the preposition "for."

Lucy and me.

(All the eyes are wrong, all the mees are right. Objective case.)

Knock-knock.
--- Who's there?
Objective case.
--- Objective case who?
WRONG! Objective case whom!

it is I

Lucy and I. I believe at least.

depends

I is correct, Lucy and I not Lucy and me

"I" though most people use "Me".

Depends on the context

Both of them are OK in this sentence.

The formal one to use is "It's important for Lucy and I to be honest"

I think you're confused because people usually say it as "It's important for me and Lucy to be honest"

Both are correct actually.. Though I think if you're like just talking to someone, you should use the second one. The first one is for when you are writing an essay and whatnot.

I

It is "I" :)

I

"I"

It's important for Lucy and I to be honest