Its the only letter that doesn't follow the naming conventions of the rest if the alphabet, i.e. it has 3 syllables as opposed to all other letters' 1, and it doesn't have a sound it makes in its name. Additionally, I've heard that "q" is the newest letter in the alphabet, yet "w" uses the presumably more contemporary word "double"
The letter Q has been around since the conception of the alphabet, derived in Latin from the regional Greek letter Koppa, itself from the Phoenician alphabet [1]. It was present in the Monk Byrhtferth's documentation of the alphabet in his Manual (1011), although was rarely used until the influence of French and French loanwords following the Norman Conquest.
The Letter W is just that - double U (or V, since they weren't distinct letters for a long while). The sound was originally represented in Old English by a different letter (wynn, ƿ) borrowed from the old runic alphabet called the Futhark, but this was replaced by vv or uu around 1300 [2]. By this point we're firmly in Middle English territory. Writing during this time was full of ligatures, letter combinations and abbreviations to save space, so the transition to fully fledged letter is quite gradual. Caxton's printing press contained a block type for the letter w, as can be seen in the famous anecdote about egges and eyren printed in 1490.
[1] Edward Maunde Thompson, An Introduction to Greek and Latin Palaeography (1912)
[2] Dennis Freeborn, From Old English to Standard English (1998)