Okay, here goes...

In my opinion, your choice here really depends on how you want to translate "Aufgriffsrecht" in this particular context, i.e. in the "Articles of Association" (="Gesellschaftsvertrag") of a German GmbH (similar to a "limited liability company").

You've provided excellent context, seeing as how your Gesellschaftsvertrag actually can be found quite easily with Google - it looks like a template of some kind.

If we take a look at where the word "Aufgriffsrecht" appears, we immediately notice that it is used together with "Vorkaufsrecht," so we can strike all of the suggested translations that might be misconstrued in that direction, i.e. "pre(-)emptive right of purchase," "pre(-)emption rights," etc., though the two are certainly related (see below).

Looking further to footnote 16 (on page 12) we find a reasonably concise description of what an "Aufgriffsrecht" is: It is basically the same as a "Vorkaufsrecht," i.e. the preemptive right to purchase the shares of a departing shareholder (or partner). But the "Aufgriffsrecht" is the right of the remaining shareholders to effect the transfer, and not just the sale, of the shares of a departing shareholder.

To make a long story short (here), I recommend reading the following document, starting on page 14 (p. 15 of the Pdf), and in particular pages 16 (p. 17 of the Pdf) and 19 (p. 20 of the Pdf), which describes pretty much the same situation that appears in your Gesellschaftsvertrag (cf. http://www.lkshields.ie/htmdocs/publications/pub071.pdf ).

:-)