They would like an English name and prefer shortened names as formal name. For example, they are Rebecca "Becky" and Gareth "Gary" and find it frustrating to have to correct people. They would prefer to have a name where the nickname is the given name, or a nickname-proof name.
They have their girls name set -- Annie Kate -- but the boys are proving more difficult to agree on.
They had agreed on Teddy George for a boy, but now Teddy is in the top 100 and rising, that has put Becky off (Gary would still like to use it).
Becky likes Hugh or Hugo; Gary likes Ralph but neither are keen on each other's choice. A relative has a little boy named Albie which, while they wouldn't use, they both really like, and feels fits the style of name they are looking for: not too outlandish, but not common.
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Good news, Becky. It looks like you are nearly there. You both seem to have a lot of common-ground style-wise and the fact that you have already chosen two choices is very positive.
For what it is worth, I think Teddy is a great option and one you should still consider. I understand that the rise of Teddy worries you, but it isn't super popular yet. It ranked #86 in 2013 which breaks down as one boy for every 500 born that year. Even the #1 name, Oliver, only works out at 10 boys in every 500, so, were Teddy to rise more, it's popularity won't necessarily mean an epidemic.
Monty -- Once Monty was the traditional diminutives for proper Victorian gentlemen Montague and Montgomery, but it has long been used as a standalone name. It has the same cute-fusty vibe as Albie, but is still very underused at #375.
Barney -- I was going to suggest Barnaby, but then it occurred to me that a Barnaby could most likely be shortened to Barney -- so why not got for just that. Barney is a dapper turn-of-the-century gem, which has been given a wider appeal thanks to How I Met Your Mother. It sits under the radar at #382.
Jasper -- Jasper isn't a shortened name, but it is fairly nickname-proof. It's a quirky vintage choice with heaps of history and modern appeal.
Rufus -- Like Jasper, Rufus is also fairly nickname-proof. It makes for a cute name on a child yet dashing on an adult.
Reggie -- Reggie is a retro short-form with cool young(ish) namesakes like Reggie Yates. It is a fashionable choice, but has not cracked the top 100, ranking much the same as Albie.
Fletcher -- One of the most dapper surname choices, Fletcher is an old-time favourite with a modern edge.
Sidney -- Vintage Stanley is already in the top 100, but his 19th century brother hasn't yet reached those heights. It has risen near the top 200 in recent years, making it a fashionable yet uncommon choice.
Milo -- Can a name be sturdy and adorable at the same time? I think Milo fits the bill. Ranking just below Albie, Milo is reconcilable but not too popular. They spelling Mylo is even less common.
Rupert -- Another name in the golden spot between 100-300: recognisable -- fashionable even -- but not common. Rupert has been given a revamp in recent years as people turn to dignified Edwardian choices and memories of Rupert the Bear fade.
Ellis -- A medieval form of Elijah that has a sleek and sophisticated modern appeal.
Bertie -- Bertie was once a staple of Victorian nomenclature, used for favourites Albert, Herbert, Hubert, Gilbert, Bertram and more. No it is used freely as a friendly but uncommon standalone name. Similar sounding Bernie is an even more uncommon retro choice.
Chester -- A great under-the-radar vintage choice. Pam and Linda of nameberry describe it perfectly as "a comfortable, little-used teddy-bear of a name that sounds both quirky and cuddly."
Winston -- You can't more quintessentially British than affable Winston. Stylish celebs Billie Piper and Laurence Fox, and Idris Elba have chosen the name for their sons, so it's staggering that it is only at #882.
Digby -- Delightfully dapper Digby has been resurrected from total obscurity to rare but stylish choice.
Felix -- A saintly upbeat name with a pleasingly happy meaning. Felix has breached a little into the top 100, but still remains rare outside of London and the South East.
Angus -- We'll chalk this one down as a traditional "British" choice. Angus is familiar to all without being too popular, and has plenty of old-school charm.
Wilbur -- Wilfred is a rising fashionable vintage gem. Wilbur is his clunkier yet adorable brother that has also risen from total non-use in recent decades to #730. Not common, but a definitely a hipster dream.
Eric -- Solid, steady but also charming. It currently sits in the #100-#300 golden spot.
Morris -- This is definitely a wildcard. Morris has been gaining more attention recently, but does not even crack the top 1000. It does have a delightful grandpa-chic to it though that puts it ahead of the curve.
Best of luck in your search for the perfect name. Please keep us updated when you decide.