The free online dictionary Wikipedia is called ウィキペディア in kana.

Why is it not called ワィキペディア? Which string of kana provides a pronunciation
closer to the English word?
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katakana loanwords kana
Why is "Wikipedia" written as ウィキペディア instead of ワィキペディア?
ウィ is the standard way of transcribing [wi] or [wɪ]. Similarly ウェ is used for
[wɛ] (for example _website_ → ウェブサイト) and ウォ for [wɒ] or [wɔ] (for example
_wombat_ → ウォンバット or _walkman_ ウォークマン).

Here ウ is used to emulate the [w] sound and ィ is a small _kana_ , indicating
the vowel. The small ィ also makes ウィ into a digraph (same length as single
full-sized _kana_ ).

This is similar to ファ フィ フェ フォ which are used to transcribe/transliterate
words containing F, where フ is used to emulate the [f] sound (which actually
becomes [ɸ]) and the small _kana_ ァィェォ indicates the vowel, giving for example
ファ [ɸa].

It doesn't make sense to use ワ here and it's actually not clear how ワィ should
be pronounced (small _kana_ aren't used with full-sized _kana_ from the ア row,
カサタナハマヤラワ).

See for example the _Extended katakana_ section in
<https://en.wikipedia.org/wiki/Transcription_into_Japanese>.

* * *

A closer transcription of the English //ˌwɪkɪˈpiːdɪə// would be ウィキ **ピー**
ディア, but transcription into Japanese is often a compromise between spelling
and pronunciation.
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Rather than providing a pronunciation closer to the English word, it provides
a description of the Japanese pronunciation.

The Japanese W sound is essentially spoken by moving from a うto a second
vowel. I. E. わ starts as if you are going to say う and then your tongue moves
into position to say あ. This is different from English W which uses more
rounded lips and is hence also why Japanese people have trouble saying words
like woman, wood, wool.

ウィ is actually a more accurate representation of how Japanese people would
pronounce this word.

On the other hand, think about how romaji is used to represent Japanese words.

  * Fuji? Barely an f sound in reality. 
  * Ra ri ru re ro? Definitely not at r. 
  * The n in ten'in 店員 or sen'in 千円? Not an n at all but actually a nasalised vowel.

But these are more familiar sounds to English speakers and provide a way for
them to speak within the realm of English phonotactics.

The same thing happens even between English and German.

  * Brezel - > pretzel
  * München - > Munich
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Apparently, even historical kana spelling used ウ rather than ワ, as in ウヰスキー
(still kept in company names such as
[ニッカウヰスキー](https://ja.wikipedia.org/wiki/%E3%83%8B%E3%83%83%E3%82%AB%E3%82%A6%E3%83%B0%E3%82%B9%E3%82%AD%E3%83%BC).
So where would the usage of ワ come from?

As you explicitly refer to Wikipedia, I found a couple of discussions on the
kana displayed in the Wikipedia logo, such as
[this](https://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Archives/Language/February_2006#Obsolete_Japanese_characters)
(from 2006):

> The current Wikipedia logo originally contained the character ヰ, obviously a
> misinterpretation of the Japanese spelling of the word wikipedia, and was
> shortly after changed to what looks like クィ, which is a nonsensical
> character combination and was probably another effort at spelling wikipedia
> in Japanese with a ワィ, which is also nonsensical. (The correct character-
> combo should be ウィ.)
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The other answers have pointed out that ウィ is the preferred kana combination,
which also happens to be the standard suggestion if I type `wi` on my Japanese
keyboard in kana mode. This answer will attempt to state why ワィ would be a bad
choice.

The most common kana to be followed by small kana are those of the /i/ series
used together with ゃゅょ to form palatalised consonants. Historically, the next
common would be く or ぐ followed by a /wa/-series small kana (わゑゐ; I am not
sure if を was used for this purpose, too) to denote a labialised consonant
which appeared and mostly disappeared in [Middle
Japanese](https://en.wikipedia.org/wiki/Late_Middle_Japanese) – [Kwansei
Gakuin University](https://en.wikipedia.org/wiki/Kwansei_Gakuin_University)
being one of the last remnants of this and still sometimes spelt くゎんせい in
hiragana depending on the intended pronunciation.

The choice of /i/ and /u/-series is not arbitrary because the vowel sound
these series have is closest to the semivowel that is intended: /i/ is the
closest vowel to the semivowel /j/ and likewise /u/ is the closest vowel to
the semivowel /w/.

In modern times, further varieties of small kana have been added to accomodate
the pronunciation of (often English) loanwords which do not fit into common
Japanese phonology. Here typically, the first kana provides a consonant sound
while its vowel is to be disregarded while the small kana provides the actual
vowel. Sometimes, as in ファ/フィ/フェ/フォ, there is only one consonant kana to
choose from. In other times, there are many such kana yet only the one with
the closest vowel is used: thus, we see ティ and トゥ for /ti/ and /tu/ but not テゥ
or トィ or any variant with タ. Using the closest vowel provides the additional
benefit that if you happen to misrender or misread the small vowel as a
standard-sized one, the resulting pronunciation still approximates the
intended one.

For /wi/, the kana sequence ワィ might suggest itself based on the ‘there is
only one consonant choice’ rule – but then note that English _w_ like its
Japanese counterpart isn’t actually a full consonant but a semivowel. This is
reflected in there being no traditional kana for /wu/ (or /yi/) although some
[were assigned in the Meiji period](https://en.wikipedia.org/wiki/Hiragana#Wu)
and also by the use of /wa/ series kana to signify labialisation where present
as outlined above. Therefore, instead of invoking a /wi/ style pronunciation
it would be much more obvious to use this sequence of kana to represent a
/wai/ style pronunciation. Indeed, the sequence ワイ (using a full-size イ) is
present in a number of loanwords such as ワイン (wine) or ワイヤレス (wireless) where
the English letter combination _wi_ is pronounced as a /wai/ diphthong.
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