If Lwaxana Troi is
a daughter of the Fifth House of Betazed, the Holder of the Sacred Chalice of Rixx, and Heir to the Holy Rings of Betazed
As Betazed is a matriarchal society with title and position being passed down the female line, wouldn't Deanna have had title and position in Betazed society as Lwaxana's oldest daughter?
As Majel Barret (and hence, Lwaxana) survived well past the end of stories involving TNG characters, I'm not sure we'll ever get a canonical answer to this question.
However, yes, given that we get the strong impression of a matriarchal, matrilineal Betazoid culture, it would seem that, unless there's an older daughter of Lwaxana we don't know about (given her marital history and that we don't really know the life span of Betazoids, I wouldn't be at all surprised), Deanna would be heir to all of those titles.
Whether she'd actually want them is another story, of course, and she might well go out of her way to find some way to disqualify herself so that they could pass to someone who cared about them more than she seems to! :-)
As far as the show is concerned, Lwaxana's position and title were mentioned on numerous occasions, in both TNG ("Haven", "Half a Life", "Ménage à Troi") and DS9 ("The Forsaken").
Although we don't see either of the objects mentioned, Troi's dismissal of the 'Chalice of Rixx' (named as an in-joke by the writers referring to Rick Berman's coffee cup) as an...
"old clay pot with mould growing inside it"
...would strongly suggest that these titles and objects are purely ceremonial.
Given the matriarchal nature of Betazed society these would certainly devolve to Deanna Troi on her mother's death given that she has no (living) older sisters.
In the Expanded Universe, in the book "Tales of the Dominion War", both the rings and the chalice (which are kept in the bottom of a cupboard) are destroyed during the Dominion invasion of Betazed mentioned in the DS9 episode "In the Pale Moonlight". This would leave Troi as merely the heir to the matriarchy of the Fifth House of Betazed.
