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I note that some people like to write the inverse hyperbolic functions not with the prefix "arc" (like regular inverse trigonometric functions), but rather "ar". This is because the prefix "arc" (for arcus) is misleading, because unlike regular trigonometric functions, they are not used to find lengths (inverse trigonometric functions can be used to find arc length of ellipses like $x^2+y^2=1$) but rather find area of a sector of the unit hyperbola.

Which version should be preferred? $\operatorname{arsinh}$ initially confused me as I did not know why "ar" was used. However, some seem to prefer this notation as it is more of an accurate description of the function.

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I think arc for arcus would be more plausibly misleading if more people actually knew about the word... –  anon Jun 12 '12 at 20:05
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My vote goes to $\operatorname{arsinh}$. I've never seen the $\operatorname{angsinh}$, but i have seen $\operatorname{argsinh}$ (arg as in argument, I suppose.) –  mrf Jun 12 '12 at 20:25
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@Argon: "ang" for "angle", as in "angle whose sine is...". As Americo writes, you also often see the prefix "arg" for argument. In Mexico, as I remember (which may be inaccurate), the common names were angsin and argsin; arcsin was not very common. –  Arturo Magidin Jun 12 '12 at 20:33
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@AdamRubinson: The confusion is that the convention breaks the usual convention for trignometric/hyperbolic functions that $\sin^n(x)$ means $\left(\sin(x)\right)^n$. So the convention would have to be stated as: $$\sin^n(x) = \left\{\begin{array}{ll}(\sin x)^n&\text{if }n\neq -1;\\ \arcsin(x)\text{ (or }\sin^{-1}(x)\text{)}&\text{if }n=-1.\end{array}\right.$$Granted, this is not an unsurmountable problem, but it is at least mildly annoying. –  Arturo Magidin Jun 12 '12 at 21:37
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@RahulNarain Here is an integration formula I copy from a book: "$\int \frac{\sin x}{\cos^n x}\,dx = \frac{1}{(n-1)\cos^{n-1} x}$ (if $n\ne 1$)". The values $n=0,-1,\dots$ are not excluded. –  user31373 Jun 12 '12 at 23:55

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Which version should be preferred?

This is easy: the version you prefer. One of the most important lessons in math is to learn to become confident in your own use of the language. Math requires coming up with new symbolism all the time. Functions and variables need names, properties need names, operations need symbols; every new object you work with, you need to be confident enough to own it as your own and look at it however you want to. Start with knowing that you can work with whatever form makes most sense to you.

When you want to communicate a result, use some common term and don't purposely be obscure. But even then, it is extremely common to see that a paper summary uses a full, common term, that internally becomes something specific to the author and their own preference. There is usually an internal logic to such choices (and if there is, much the better). Which gives another reason why it's important to be flexible with one's choice of symbols: it will help you read other people's papers.

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