Learn English – Abbreviating names that start with a vowel

The use of the apostrophe to denote an abbreviation (more accurately, to denote missing letters) is known as an "apologetic apostrophe". It is recommended by Eats, Shoots and Leaves that apologetic apostrophes be used only for "novel" abbreviations. They should not be used for abbreviations that have become common words in their own right (fridge, nuke, phone). The exceptional case is for those abbreviations more commonly known as "contractions"; "I've", "you'll", "don't", etc., where the use of the apostrophe became common along with the word.

So, short answer, it depends on your audience; those who have never seen the term before would probably understand it better with the apostrophe, while those who use the term in speech would get it faster without. The fact that it's audience-dependent would lead me to say that the abbreviated term is jargon and should be avoided when writing for a general audience. However, given that, the solution becomes simple; drop the apostrophe, because if you're using it at all, it's common to your audience.

Shortened versions of surnames (last names) are often used as nicknames in English-speaking countries, but normally by people on very familiar terms with the person and therefore not in quite the same way as Robert might be shortened to Bob or Steven to Steve.

In my experience (predominantly British English) longer surnames such as Robinson and Brightman might be shortened to Robbo or Brighty or even extended from one sylable to two with the addition of an "o" or "y", e.g. Brown to Browny.

I would emphasise that this isn't something that would happen outside relatively close circles of friends, teammates or colleagues.