In this instance I feel the old madwoman is being fawning and servile.
I can't recall, was she aware earlier that Esther and co were the wards in Jarndyce, when she was being condescending? Of course the explanation for that (if she was) could be that she was channelling a more superior persona :-)

Netla, I think the statement "Charmed, I'm sure," or "Extremely honored, I'm sure," can have several meanings depending on context. As I haven't read the Dickens in question, I'm not sure which way that statement is meant, though fawning as Colleen said would make perfect sense.

From the context of the older books I've read,I think the phrase can stand in several ways:

As a banal social greeting (I don't know you, but you are apparently an equal and come recommended, so I'm happy to meet you);

As a condescending greeting or sarcastic acknowledgement (You are so far beneath me it's laughable I'm even speaking to you, so I'll sarcastically use this very polite phrase simply to make you uncomfortable, which you should be);

As a way of lulling someone into trust (Look how kind I am, speaking to you as an equal and greeting you with polite speech. Now you can be hopeful that I'll be kind/helpful/reasonable, which I probably won't be but you don't know that yet);

As a way of fawning over or attempting to placate someone (You don't know what to make of me, but look, I'm saying I'm charmed to meet you, therefore indicating my wish to be helpful to you, so please don't ignore me).

I've probably over-analyzed it, but It is amazing how one simple phrase can mean a multitude of things given tone of voice and context.

netla, as a true Native I would suggest that "Charmed, I'm sure" in this context is said by someone who is not quite in command of the correct etiquette in these situations. Is the speaker Miss Flite?
As Colleen says she is being servile but, I think with pretentious overtones . . . and probably thinks that might be how a Duchess would speak to someone.
When she goes on to say . . " very much indebted for the omen it is surely a malapropism and just adds to the fact that she doesn't know the correct words to use.
An English eg today would be to say "How do you do?" (in a formal setting when introduced to a third party) to which the answer should always be "How do you do?" No one really wants to know that you have a slight cold or sprained your ankle last week.
In fact my grandmother would probably have considered "Charmed I'm sure" to be bordering on the common. It can be a positive minefield out there.

Thank you all for the insight. This is clearly one of those phrases one should avoid in order not to accidentally cause offence.

The speaker in the example is indeed Miss Flite, although her name is unknown at this point.

Colleen, I think you're right - she does seem to be simpering at them at this point, although earlier she had come across as being somewhat haughty.

You asked if she was she aware earlier that Esther and co were the wards in Jarndyce, when she was being condescending.
Yes, she knew from the start. I think perhaps she just swings between pitying them and being condescending and then remembering what a "great" case they are involved in and fawning on them.

Veer, the omen in this case is that she thinks they will bring her luck, and indeed that word could be used in place of "omen" in some instances. Here she uses the word in a way that comes across as ironic, given the situation:
"... It is a good omen for youth, and hope, and beauty when they find themselves in this place, and don't know what's to come of it."

Then meeting her is a good omen for them: "... I shall confer estates on both - which is not being troublesome, I trust? I expect a judgment. Shortly. On the Day of Judgment. This is a good omen for you. Accept my blessing!"

And then it's a good omen for her to meet them: "... Pray come and see my lodging. It will be a good omen for me."

"Charmed, I'm sure." is a rather back street , not quite polished version of " I am pleased to meet you."

Perhaps another way to say " How lovely to make your acquaintance" or as the French might say "Enchante".

I don't think that we have to put meaning to every word.......it's the general feeling which is expressed.

I've always loved using the Marilyn Monroe response
"Thank you ever so."
It's just a bit quirky and not the hackneyed " Thanks."

Vee, yes, a minefield indeed! A few words at the beginning of an acquaintance can pigeon-hole the speaker for the listener in seconds!

Ann, Ms Monroe might have got away with "Thank you ever so much" but we English speakers know that the 'ever' word would be social death and has to be replaced by 'very'!
My 'Southern' aunt used the word 'elegant' in ways I was not familiar with. For eg she would describe a meal as elegant meaning well-cooked/tasty. Whereas were I to use the word at all I might have said the dining room or the table setting was elegant. It could just be that the meaning has changed between UK/US over the years.

Vee, I know that "ever" is frowned on but I must admit that, when I order a book from an English firm and get the informal "thank you ever so much" emailed reply, I smile. Ever so much more friendly than a formal one.
It makes me think that I have been served by a young and enthusiastic girl shop assistant! I can even visualise her...

Oh !
Ms. Monroe, what have you done to me !?

I don't think I have ever heard someone say to me "Charmed, I'm sure." Possibly, it is because no one has ever been charmed....or I look like a person who would consider it sarcasm and would clobber them with my purse which contains some doorstopper of a library book. For some reason Hugh Grant comes to mind for that phrase.
And whether someone uses ever in a "thank you" doesn't matter to me. I just appreciate the courtesy. People are not even making eye contact anymore and are oblivious to social niceties because they are too busy texting. We have pedestrians that are getting hit by cars because they are texting as they are walking and step off curbs without looking....
I blame it all on twitter and facebook. In fact one day, people won't even say thank you, they will text it. If you are lucky.

And if you say thank you, the response will be "No problem." Drives me nuts (nuttier?).