It isn't just a matter of who gets more hits on Google. In 2000, the OpenOffice name was not trademarkable worldwide (and probably still is not) without the .org. OTOH, the ASF doesn't normally seek to register all it's trademarks worldwide because of cost (in the US, first use of a mark is significant enough)...but if Oracle is transferring it's trademarks to the ASF (and if it's the worldwide trademark, then it's worth about $350,000 USD), then we're back to dealing with the .org, I'm afraid.
Danese Kay Shenk wrote: -Yeah, once upon a time, when you did a search on "Open Office", the *other* office suite, whoever owned this brand, came up (like maybe 6+ years ago). Now it doesn't, but we should ascertain the status or history of this before changing the product/brand name I think, just to be safe.