“Shaw” → “Shavian” – why “v”?

It is said that Shaw disliked the adjective "Shawian" (which does look and sound awkward) and proposed to invent a new one.

He Latinized (sort of) his surname, from Shaw to Shavius (there is no "w" in Latin, while "u" is spelled as "v").

Shavius naturally lends itself to Shavian.

At least that is the explanation given in the Dictionary of Eponyms by Martin Manser, which is available online in fragments (and also in its complete form for a fee).

Frank McNally in the Irish Times says it's a mystery, but he thinks it may because of rhoticity Shavian lather – An Irishman’s Diary on descriptive derivatives, while Morton S Freeman alleges that Shaw invented it himself, by 'Latinising' his surname. A New Dictionary of Eponyms

Just throwing another one into the pot..

According to 'the surname database' Shaw has several different modern-day versions:

the modern forms of the surname range from Shaw(e), Shay and Shay(e)s to Shave(s) and Shafe.

Accordingly Shave → Shavian would still be an etymological derivation as I understand it.

The article claims that Shaw derives from 'sceaga'.

As a topographical name, Shaw was used for someone who lived by a copse, wood, or thicket, derived from the Old English pre 7th Century "sceaga", copse, small wood. As a locational surname, Shaw is derived from any one of the numerous small places names Shaw, from the Old English "sceaga", such as those in Berkshire, Lancashire, and Wiltshire.

Throughout the centuries, surnames in every country have continued to "develop" often leading to astonishing variants of the original spelling