Source: https://www.telegraph.co.uk/royal-family/2019/04/29/royal-baby-name-odds-title-meghan-markle-prince-harry/
Girls
1. Diana 4/1
2. Victoria 7/1
3. Alice 12/1
4. Grace 12/1
5. Isabella / Isobella 12/1
6. Elizabeth 14/1
7. Alexandra 20/1
8. Harriett 20/1
9. Rose 20/1
10. April 25/1
Boys
1. Arthur 16/1
2. James 16/1
3. Edward 25/1
4. Albert 33/1
5. Alexander 33/1
6. Christopher 33/1
7. Daniel 33/1
8. Henry 33/1
9. Phillip 33/1
10. Joseph 40/1
My Opinions
For girls I'd like to throw Diana, Grace, Isabella, Harriett, Rose AND April out of consideration. IF a month name were to be used, May would be better as Harry's great great grandmother Mary of Teck was known as May, and we're nearly in May now. That leaves Victoria, Alice, Elizabeth and Alexandra from the top bets... Harriet was on my list of preferred names but not with two t's and because it was a great grandmother's name.
For the boys names I'll throw out Edward, Albert, Christopher, Daniel, Henry and Joseph. I think it very unlikely the baby will be named Henry as that's Harry's proper name and that is rather out of fashion of late. That leaves Arthur, James, Alexander and Phillip. Arthur Alexander was one of my top picks if I'd ever had a boy.
A good few people have suggested the idea of twins in which case using Elizabeth and Phillip would be rather nice but I don't think twins are especially likely as Meghan is allegedly planning a home birth and has apparently gone to full-term. Not that these things CAN'T happen but an older first time mum having Royal twins at home seems a stretch too far and an early labour is statistically more likely.
Obviously many other names have been suggested including a nod to Meghan's mother, Doria...I wonder if they might splice Doria and Diana like Renesmee in The Twilight Saga who was named after Bella's mum Renee and Edward's adoptive mum Esme. Daria? Dorana? Hmmm...
Unlisted names I think might make an appearance are Helena and Leopold.
My Prediction
Names in order of preference and using the Royal family's tendency to give four Christian names:
Girl: Helena Alice Doriana Victoria
Boy: Arthur Alexander Leopold James
Now just for time to tell.
EDIT - 03/05/19
Royal baby name odds, from William Hill, updated 03/05/19
Source: https://www.telegraph.co.uk/royal-family/2019/05/03/royal-baby-name-odds-meghan-markle-prince-harry-boy-girl/
Girls
1. Diana 4/1 (no change)
2. Allegra 8/1 (new)
3. Grace 8/1 (up from #4 at 12/1)
4. Alice 12/1 (down from #3, same odds)
5. Isabella / Isobella 12/1 (no change)
6. Victoria 14/1 (down from #2 at 7/1)
7. Elizabeth 20/1 (down from #6 at 14/1)
8. Alexandra 20/1 (down from #7, same odds)
9. April 25/1 (up from #10, same odds)
10. Catherine 25/1 (new)
Harriett and Rose have left the top 10 in exchange for Allegra (rather like it) and Catherine (good name but the same as Harry's sister in law? That'd be a tad odd...although it would put those stupid feud rumours to bed).
I've started thinking of Eleanor as a good, classic Royal name long overdue for a comeback. I am still VERY much of the opinion that if they don't use Doriana for a daughter they are very very wrong indeed. So my updated girls' name is now Eleanor Doriana Victoria - also shifting to the three christian names pattern adopted by William & Catherine.
Boys
1. Arthur 12/1 (up from 16/1)
2. James 16/1 (no change)
3. Albert 20/1 (up from #4 at 33/1)
4. Edward 33/1 (down from #3 at 25/1)
5. Alexander 33/1 (no change)
6. Christopher 33/1 (no change)
7. Daniel 33/1 (no change)
8. Henry 33/1 (no change)
9. Oliver 40/1 (new)
10. Phillip 40/1 (down from #9 at 33/1)
So only one new entry - Oliver has been a very popular name in the UK for a long time but it's association with Oliver Cromwell makes me point and laugh at the idea of it being used as a Royal baby name.
I'm still quite happy with my boys' name guess but if I was switching to a 3 name combo I'm not sure which I'd drop... Arthur Alexander Leopold, Arthur Leopold James? Hmmm, wonder if either of them are James Arthur fans?! Maybe Harry is more of an Ed Sheeran fan...remember the commemorative plate incident (source: https://planetradio.co.uk/heat/entertainment/celebrity/ed-sheeran-prince-harry-mistake-plate/)? Maybe for a 2nd guess I'll go Arthur Leopold Edward ;)
CONCLUSION
#BabySussex was a BOY - delivered safely on Monday 6th May 2019 at 05:26 BST and weighing 7lbs 3oz. It was announced on the 8th that Harry and Meghan are naming their son ARCHIE HARRISON MOUNTBATTEN-WINDSOR, he does not have a title at this time. Basically all the top guesses were wrong - from the expectation of a girl to the two or three expected middle names. No idea where Archie might've appeared in the betting, if indeed it was suggested at all. On the plus side maybe their naming a potential future daughter 'Doriana' isn't so unlikely after all!
Someone on Twitter disagreed with my bet on 'Arthur' (despite it being the top of the bookies shortlist by that time) "as Pippa just had a son she named Arthur" ...Pippa being Catherine, Duchess of Cambridge's sister Philippa Matthews nee Middleton. As such she is no relation to Harry & Meghan so her choice of baby name is fairly irrelevant.
If such a thing were a factor it obviously means very little to the Royals as evidenced by Prince Louis of Cambridge, William & Catherine's third child. Louis Arthur Charles was born a little over a year ago. Not only does he share a name with older brother Prince George (George Alexander Louis) and his father (William Arthur Philip Louis) but he also has a STEP COUSIN of the same name! Prince William and Prince Harry's stepbrother Tom Parker-Bowles has two children LOLA (b. 2007) and FREDDY (b. 2010) and their stepsister Laura Lopes has three children ELIZA (b. 2007) and twins GUS and LOUIS (b. 2009) - all these step cousins having been born since Prince Charles marriage to their grandmother Camilla in 2005.
I very much like the name 'Archie' but I'm not a huge fan of 'Harrison' because - and you can laugh at this - I don't like surnames used as Christian names. Yes, I know 'Heggie' is a surname and yes, I chose it myself. Surnames are good as gender neutral names though...
Speaking of this a good few people liked the name 'Spencer' - after Princess Diana's maiden name - for a child of either gender. Despite her enduring popularity I can't imagine it being used as a Royal baby forename, a middle name perhaps, but overall I would rather William and Harry looked to the future in their own families. Not only that but let's not lose sight of the fact that Catherine and Meghan both have their own families and connections - it is not only William and Harry who might want to commemorate loved ones in their naming of their children.