This is just my observation, but it seems to be that the Society (JW's) generally favor those Bible translations that are friendly to their theology. For example, the JW's favor Moffat and Goodspeed for their renderings of John 1:1c and you will occasionally see these translations in Witness homes.
The JW's actually own the publishing rights to Stephen Byington's translation and apparently hold it in high regard. (Probably for it's use of "Jehovah" possibly coupled his favorable review of the NWT) I've also been told that the JW's attempted to purchase the rights to the Concordant Literal Version probably (And this is more speculation on my part) because the CLV's renderings of most of the traditional trinitarian proof-texts are favorable to JW theology. (But overall, it is one godawful translation if you ask me.)
One other translation that the JW's hold the publishing rights to is Benjamin Wilson's Emphatic Diaglott, an interlinear that appeals to them because parousia is rendered as "presence." However the last time I checked, the JW's no longer print this work (Or any other interlinear for that matter.)
The desire to find and promote translations favorably to their theology also has apparently led them astray at times, as their 20 year period of quoting from Johannes Grebers translation of the New Testament is now a source of considerable embarassment today.