It is common to abbreviate a loanword to four morae, and when the original loanword is a compound word consisting of two components, then it is common to do so by taking the first two morae from each component. Sometimes sokuon (little tsu: ッ) and chōon (ー) are skipped when counting two morae for this purpose.
- リモートコントロール (remote control) or リモートコントローラー (remote controller) → リモコン
- パーソナルコンピューター (personal computer) → パソコン
Sometimes the final sokuon or chōon after abbreviating is dropped, and this explains ブラピ in Jesse Good’s answer and ハピバ in Tim’s comment on Jesse Good’s answer.
But not every loanword is abbreviated like this, as Jesse Good already wrote.
This pattern is not really specific to loanwords. I have seen this pattern applied to the title of comics, video games, and TV programs. In some sense, even abbreviations such as [外国]{がいこく}[為替]{かわせ} to [外為]{がいため} may be viewed as an instance of the same pattern.
By the way, I do not think that ドンマイ is an abbreviation of something like ドントマインド. It seems to me more like a transcription of the pronunciation of the phrase “Don’t mind” heard by Japanese speakers.