What comes to your mind when you hear these names?

Virginia I would assume an American.

Livia and Lavina an middle class English name.

Clelia I would assume a spelling mistake before I had read through this thread.

They sound like quadruplets in the latest Jilly Cooper.

Livia. IMHO it looks as if you missed the O from Olivia
Clelia. Never seen this name before , the fact that you have to explain how it's pronounced makes it a no , from me
Lavinia. I like this , unusual now , but pretty
Virginia. Again , I like this , but if for my daughter I'd worry she'd be teased about being a virgin ( not that theres anything wrong with being a virgin )

So. Of the four , I prefer Lavinia

Livia - Someone missed the O from Olivia
Clelia - A mis-spelling of Celia and a tricky one to pronounce
Lavinia - Upper middle class, not current but also not majorly dated
Virginia - The US state also Virgin.

I haven't heard of the first two names. I think Lavinia did some horrible murder in an Agatha Christie book and I'm not keen on Virginia, although I like Ginny as a diminutive.

Livia - I love this. Very Roman and much nicer than Olivia
Clelia - sorry but I immediately thought of cloaca. I don't like it at all
Lavinia - I quite like this and think I could love it
Virginia - I think it's too difficult for anyone to live with

@StepOutOfLine

I'm Italian so know they all are but I only know actual Italians with the first 3 names, and anyone I've met called Virginia was English or American so I wouldn't have that same association with Virginia as I would Livia, Clelia or Lavinia.

Livia - OK but looks like the O has dropped off of Olivia

Clelia - no opinion at all really

Lavinia - Titus Adronicus' daughter who is rapped and has her tongue cut out and her her hands, chopped off

Virginia - Woolf and the Bloomsbury set

Livia - I would assume you were Italian (lovely name!)
Clelia - never heard of this. Assume it was a typo.
Lavinia - The sort of name that is fine if you’re beautiful. Otherwise, a bit frilly.
Virginia - American State. I like the shortening Ginny.

Livia - pretty. I know a lady called Livia (she's not originally from the UK so it may be more common in her home country)

Clelia - hmm not sure I like this one, sorry.

Lavinia - lovely but I'm biased (my beloved late grandmother's name)

Virginia - it's ok. I had a lovely babysitter called Virginia when I was a child.

Livia - quirky spelling of Olivia
Clelia - might be a contracepive device?
Lavinia - Lav
Virginia - tobacco

Livia - Lively
Clelia - Klingon, clitoris. The "kl" sound is not pretty.
Lavinia - Lavatories
Virginia - Initially, I think of that song that goes "Almost heaven... West Virginia". Then I think of virginity.

Livia - I like it but i think people will put an O in subconsciously and she'll spend a lifetime saying Livia, no O. Cute, sweet, pigtails. Uses Lov when she's older
Clelia - I don't like it and think people will subtract an L subconsciously, and she'll spend a lifetime saying Clelia, with an L. I can't come up with an image.
Lavinia - spoilt little madam, always wearing the finest clothes (do NOT jump in that puddle Lavinia, get away from that DOG Lavinia, no you can't go on the slide in that dress Lavinia) who's over indulged.
Virginia - sensible child who prefers reading boring books to playing and had no friends. Lives in sensible jumpers and shoes

Livia, Lavinia and Clelia, I'd assume they were Italian (if pronounced properly) and maybe a bit pretentious if not.
Virginia makes me think of Virginia Woolf

Livia - Colin Firth’s wife or someone to lazy to say Olivia
Clelia - “how the fuck do I pronounce this?”
Lavinia - Mean girl from the book “A Little Princess”. Lavatory/Latrine
Virginia - Woolf. Virginity. Virgin.

I'm writing this before reading the other answers so I can't be influenced
Livia - Roman, pretty name, maybe short for Olivia, name of a friend's child, relatively posh name but not overly so
Clelia - misspelling of Celia? If not, hard to say, unless it is pronounced some other way than it looks.
Lavinia - It does make me think of toilets I'm afraid, and also the nasty girl from A Little Princess. Buttoned up and snooty.
Virginia - Ginny from Harry Potter, the US state, Virginia Water which I think is in Surrey, and virginal. Not a name I would want to be saddled with myself.

I love Livia though.

Livia - nice Italian name, inspector Montalbano’s girlfriend
Clelia - sounds like a disfigurement
Lavinia - posh, wasn’t this the name of that Oxford student that stabbed her boyfriend?
Virginia - Woolf

Livia - dropped the ‘O’
Clelia - sounds like something from a Latin textbook and is a bit of a mouthful
Lavinia - one of the characters in a Miss Marple book, but I quite like it. It’s old fashioned but not too off the wall.
Virginia - I think of the US state, but it’s a perfectly okay name.

I like Virginia best.
I'm in Italy and have students with all 4 names.
Clelia- they've all been named after their grandmother. Which is traditional still in many families. Ditto Lavinia.

Livia - Nivea, or that you meant Olivia and it's a typo
Clelia - Again, I'd assume this was a type and you meant Cecilia or Celia
Lavinia - Nice name. I might assume she was black
Virginia - it's ok, obviously reminds me of virginity and America

Livia - short version of Olivia (I know it isn't, but there are so many Olivias around called Liv or Livia for short that it will be assumed)

Clelia - really dislike the Clel feels like too many Ls.

Lavinia - fine, no real associations.

Virginia - I always think of the US state, for absolutely no reason. As it is a name based on the word Virgin (as in the virgin Queen - names after Elizabeth 1) I think I woudl feel cautious about using it. Although I suspect most go by Ginny as a short name

First three - Ancient Rome, blood and gore.
Virginia - old American lady.

Livia - wife of Augustus; a friend of mine; a baby I met recently!

Clelia - Roman, can't think that I know any

Lavinia - the Aeneid

Virginia - someone I know in her 40s

I think unfortunately Clelia is a bit difficult in English. Claudia is nice though? Julia and Gaia are also a bit nicer as Roman names. I also know an Ilaria and an Eliana.

@LivLavender ooh that’s actually lovely! Will adjust my choice to that one aha- my initial thought would have been “Klee-lya” which is much less nice for whatever reason. Thanks for explaining!