Could Harry and Meghan's daughter become Duchess of Sussex?

No and Yes. Anne’s kids had no automatic right to any titles, unlike her nieces and nephews. The Queen can grant anyone a title she likes, and was minded to do so, as these were her first grandkids. It was Anne who asked her not to.

An interesting post script. Zara carries out Royal duties on occasion. A few years ago when there was a distinct shortage of women in the Royal Family, there was some talk of giving her a title. She already has an MBE, earned due to her equestrian career.
With the arrival of Kate and now Meghan, that’s moot now.

Her husband declined the Queen’s offer of a earldom (customary for male commoners who marry princesses of blood royal) and when Anne was pregnant with her first child the Queen offered to make her children princes and princesses by letters patent, but Anne declined.

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Does the Queen have free choice in this sort of thing, or does she have to do what the government “suggests”?

My understanding is that honours for members of the Royal Family are her personal discretion, as are a few of the orders, but other than that, it’s on the advice of the Government.

That is not correct. France appears to have solidified “Salic law” because of a desire to steer the sucession away from people they did not want .

Due to the marraige of daughters off to foreign royals, france had the problem that the the foreign royal looked like becoming monarch . Eg the daughter of the spanish queen might have been next to rule France, even if the spanish queen herself was excluded by being spanish queen.
A number of european kingdoms have a so called “semi-salic law”, though not by derivation from “salic law”. More correctly its “agnatic-cognatic primogeniture”, which means that there must be no male descendants elegible at all, before any female is considered . There might be variations, it might be that decisions were made without solidifying the law… a monarch often nominated their successor bearing in mind the chance of that nomination being successful, so some might have nominated a male even though technically there was no law to prevent a female… thus it becomes hard to know if they had a system , or just perceieved one.

The entire concept of nobility and royalty, which categorizes people by status, is an affront to humanity. With that perspective, it’s not really all that sad that it’s also sexist.

I think the basic principle is that honours for services directly to her and the royal household’s operations are entirely her business, but that honours for military, public and community services more broadly go through the nominating and vetting committees reporting to the Prime Minister.

She gets to pick the members of the Order of Merit too. These days that is mostly for contributions to the sciences and the arts. It’s very much the cool kids club, restricted to 24 living members.

The monarch also has sole discretion as to who gets named to the Royal Victorian Order: Royal Victorian Order - Wikipedia

Order of Merit and Royal Victorian were the two I was thinking of.

I know a fellow who was made a member of the RVO, receiving his induction directly from the Queen in a private ceremony, for services rendered over his lifetime.

Bumped.

It’s been announced that the Duchess of Sussex is now expecting: Prince Harry and Meghan, the Duchess of Sussex, expecting their first child | CNN

The OP’s question can now be answered:

“Heirs male,” so if their first child is a daughter she will not be in line to become Duchess of Sussex.

Before anyone asks, the future offspring of the Sussexes will not automatically be princes or princesses. However, HM can make them so if she wants.

Speaking hypothetically, of course, would “heirs of his body” exclude children conceived via sperm donor, were he to prove to be infertile?

Traditionally, stepchildren and adopted children were excluded, so I would expect sperm donor children to be excluded also.

Thinking outside the box. H&M’s bun in the oven turns out to be a female. Harry dies off at some point leaving the Duchy vacant. The then current King/Queen makes this daughter Duchess of Sussex.

So sure she can.

Yes, she can. For that matter, even if Harry doesn’t die, the monarch could confer a title on Harry’s daughter.

But there is no expectation that she will do either of these things.

Thank you.

It should be noted, though, that even if she doesn’t give them titles now, on her death, assuming the Prince of Wales survives her and becomes king, any Sussex children will automatically become princes/princesses since they will then be male-line grandchildren of a reigning sovereign.

The precedent has been set with Princess Charlotte and Prince Louis. Prince Edward of Sussex and Princess Mary and Elizabeth also of Sussex, here we come.