Lots of names are short for longer names. Beth is short for Elizabeth. Nobody writes it 'beth. In English, when shortening a word by lopping off the beginning of the word, the tradition is to treat the shortened word as a full word in its own right, and to capitalize it as such.
That said, lots of people have insisted on defying orthographic conventions in the spelling of their names. Prince even changed his name to . Needless to say, people tend to find that sort of thing quite pretentious. So Tina's name is short for something: why should anyone care? The fact that she angrily shoves that in people's faces whenever they try to spell a normal name with normal rules is pretentious. Especially given that this is a spelling-only issue: orally, 'tina is equivalent to Tina and only people who have reason to write her name will even be aware of the distinction.
You could, of course, use the "correct" capitalization in the text to flag the speaker as one who is sympathetic to 'tina or not, as in the Eddings example in Lauren Ipsum's answer. But I suspect a person who insists on such an unorthodox spelling of her name is going to be unhappy a lot of the time, when she tries to enter her name into a computer, or fill out a government form, or apply for a job, etc.
My advice would be to have an alternate way of indicating that the name contains something else. For example, some people use their middle name, so John Harold Smith becomes J. Harold Smith and everyone refers to them as Harold. But anyone who writes down the full name should include the J in it, even if they don't know what it means, and Harold could routinely sign his emails, letters, etc, as J Harold Smith.
Language is primarily spoken for most people most of the time. And thus the token that identifies you typically is a sound, which then gets transcribed according to the local spelling rules. Violating those rules makes your character stick out a lot, and maybe not in a good way. And doing it so that your character can effectively constantly tell people "I know something you don't" seems incredibly annoying.