I do not think that I have ever seen 日語 for 日本語 in Japanese. Both Daijirin and Daijisen list the word 日語 with the meaning “the Japanese language,” but Daijirin notes that the word is used in Chinese, Korean and so on. Indeed 日語 sounds like a Chinese word to me.

There are a few words which mean almost the same thing as 日本語:

• 国語 (こくご): Literally means “national language,” but it means the Japanese language. For example, classes on the Japanese language at school are called 国語, and the study of the Japanese language is called either 国語学 or 日本語学 at universities.
• 邦語 (ほうご): The same as 国語.
• 和文 (わぶん): Means “text written in Japanese.” For example, Japanese fonts are often called 和文フォント.
• 邦文 (ほうぶん): Literally means “national text,” but it is used synonymously to 和文.

However, note that it is usually called 日本語, and if you use another word, it can imply something.

Probably I should write my opinion about the logo on meta, but I will continue here. As far as the logo is concerned, honestly, I do not think that any of these replacements works. 日語 sounds like a Chinese word, and I get a contradictory impression. I feel that 国語 and 邦語 (and 邦文) have an unnecessary focus on “the national” language which does not really make sense on an international website. 和文 and 邦文 are slightly off because they mean not the language itself but the text written in Japanese. Does 日本語 with the lower left corner left blank really look bad?

ES, DE, EN are all ISO codes -- "ISO" as in "International Standards Organization" (technically, it's the "International Organization for Standardization"). These codes are used worldwide, particularly in programming and other technical contexts. In localization engineering, I've dealt with these codes for years. There's no particular value in developing a competing standard—the point of a standard is that, well, it's the standard.

That said, Japanese already has a convention for abbreviating country names in text -- usually the first kanji of the country's full name.

• 英【えい】 for the UK / English, from 英【い】吉【ぎ】利【り】[　]{す} (Igirisu, probably from Portuguese Inglês, with the spelling from Chinese)
• 露【ろ】 for Russia / Russian, from 露【ろ】西【し】亜【あ】 (Roshia, from Russian Росси́я (Rossíja), with the spelling probably partly from Chinese)
• 仏【ふ】 for France / French, from 仏【ふ】蘭【らん】西【す】 (Furansu, ultimately from French France, spelling derivation unknown)
• [独]{ど or どく} for Germany / German, from 独【ど】逸【いつ】 (Doitsu, from Dutch Duits //dœy̯ts//)
• 中【ちゅう】 for China / Chinese, from 中国【ちゅうごく】 (Chūgoku, from Chinese 中國)
• 日【にち】 for Japan / Japanese, from 日【に】本【ほん】 (Nihon)

And so on and so forth. Pretty much every country that has a kanji-based name can be abbreviated in this way. However, there are exceptions, so be sure to look things up:

• 蘭 for the Netherlands / Dutch, from 阿【お】​蘭【らん】​陀【だ】 (Oranda, from Portuguese Holanda with a silent "H", meaning "Holland" -- which historically was the leading province of the Dutch Republic and roughly corresponds to the provinces North Holland and South Holland, two of the twelve provinces that make up the present country of the Netherlands)

Update

As pointed out by Earthliŋ in the comments, sometimes Japan and Japanese are referred to by the abbreviation 和. This stretches back over a thousand years to the initial textual references to any country referred to as "Japan", when Chinese-language sources describe the country of "Wa" or 倭 (wa, literally "dwarf" in Chinese). See the 和 entry at Wiktionary for more detail. (Full disclosure: I edited that entry.)

One example of modern usage of this 和 to mean "Japanese" is this German-Japanese dictionary over on Amazon, using 独和【どくわ】 to refer to "German" and "Japanese". It's also the component used in the term 和文【わぶん】 "Japanese text", in contrast to 漢文【かんぶん】 or "Chinese text". Or in 和語【わご】 "Japanese language" in contrast to 漢語【かんご】 or "Chinese language".

Meanwhile, there is also a dictionary from the early 1600s that has been very important in discovering how Japanese sounds have changed over the centuries, called the 日葡辞書【にっぽじしょ】, using 日 to refer to "Japanese" instead.

So I think both 和 and 日 would work as one-character abbreviations for "Japanese". My subjective sense is that the two might be used in different contexts, but I do not understand the difference well enough to explain it here.

Please comment if the above does not address your question.