Elizabeth is in the process of adopting her third child -- gender unknown -- who will be a sibling for Elijah and Margaret.
Elizabeth writes: "I like first names with more than one syllable, I don’t like names that are stand-alone nicknames (i.e. if my mother named me Beth instead of Elizabeth, or I named my son Eli instead of Elijah), I also like to take into account the meaning of the name [and] I like the royal trend on the female side.
I really like popular names such as Oliver, however, I am not sure that I want to use such a popular name. I didn't realize Elijah was so popular until after we had our son and I started noticing there are quite a few little Eli's in his grade. I don't like trendy names very much either, I like a little more history with the name.
Archie ("I really like this one, but not so keen on Archibald or Archer as the full name")
Philip
Nathaniel
Augustus
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Both Elijah and Margaret have a rich heritage of use as names, though their modern popularity (#41 and #392 respectively in England and Wales 2018) sets them a little apart. For options for Baby #3, I've looked for names which are equally rich in history but manage to bridge the modern usage gap.
Alfred – At #106, Alfred is a stately and royal name which isn't too popular. It also boasts the only king of British-origin to gain the title "The Great".
Dominic– A classic name borne by saints and popes. It has a sleek Latin style and (just sitting under the top 100) an uncommon familiar.
Jonathan – A play on Nathaniel on your list, Jonathan is Biblical like Elijah (and a heroic figure at that) while feeling classically British like Margaret.
Matthias – Both a Biblical name and a royal one, not only borne by a Holy Roman Emperor but also several other European kings. At #544 it is also familiar but relatively underused.
Maximilian – A grand Roman name which has passed right through to modern royalty. At #149, it is familiar but not trendy.
Nicholas – A staple of British naming since the Middle Ages, Nicholas has been borne by saints, kings, philosophers and everything in between. Despite its enduring classic status, Nicholas hasn't been in the top 100 since 2004.
Otis – A quirky English form of royal Otto that has a dapper and debonnaire style.
Robin – A traditional English diminutive of classic Robert, Robin dates back to the Middle Ages. It's heroic but also adorably charming.
Tobias – A distinguished Biblical name which has a friendly and sprightly feel.
Vincent – A saintly and artistic name with a classic style and friendly in-built nickname Vinnie that ranks at #145.
Walter – A name with plenty of history with bearers including adventurer Sir Walter Raleigh and author Sir Walter Scott.
Wilfred – An affable Anglo-Saxon name which has persevered throughout the ages.
Alexandra – A lady-like name borne by queens and empresses.
Antonia – An illustrious Roman name borne by several members of royalty across Europe.
Bridget – Bridging the styles of Beatrice and Meredith on your list, Bridget is a name with an ancient heritage, ranking in the top 20 in England in the 16th century alongside Margaret. Bridget also counts as a royal name: Princess Bridget of York was a daughter of King Edward IV.
Caroline – A royal name borne by a remarkable British queen. It is dignified and familiar but is not at all common.
Cecily / Cecilia – A medieval staple which was borne by royal princesses and Duchess. The name is often regarded as meaning "blind", but, really, to the Romans, it signified "descendants of Caeculus" who was the son of the fire god Vulcan, born of a spark. He was given the nickname "little blind one" simply because the smoke from his father's fire caused him to squint. So the name has a rather deeper and richer meaning. It became popular in the Middle Ages thanks to St Cecilia, the patron of music.
Constance – A charming virtue name which has been in good use since the Middle Ages. The name has also been borne by many queens, princesses and other royal women.
Helena – From Roman Empress to British princess, Helena is a chic name which has an ancient heritage and rich history.
Henrietta – Another queenly name which is rich in nickname options and has the same noteworthy feel as Margaret.
Josephine – Many European queens and princesses answered to this polished and graceful name.
Louisa – Both Louisa and Louise have been borne by a number of British princesses starting from the Georgians. The name has a quaint Austen-vibe to it and ranks very close to Margaret in popularity.
Theodora – An ancient name borne by Empresses which has a vintage Victorian feel.
Vivienne – From the stately Roman Vivianana meaning "alive", Vivienne has plenty of gravitas coupled with an upbeat sound.
I hope this has been of use. Best wishes for finding the perfect name.