The question isn't "What can Open Office do for Google?", it's "What can Google do for Open Office?". A number of things come to mind when thinking of how OOo can integrate with Google. Some of them aren't likely to happen soon, but they are all possible and should be worth considering.
The most obvious, and already realized idea, is of Google providing manpower to give the software a boost. With the brains of Google's skilled engineers, there is no doubt interesting things can happen. The first order of business should probably be putting OOo on Jenny Craig. The office suite is a memory hog, and is painfully slow even on "good" machines. To be a viable alternative to MS Office there definitely needs to be some effort put into optimization.
The functionality that is most talked and speculated about is a completely online version of Open Office. Almost all the features available in the suite are technically possible using modern browsers which are capable of utilizing rich web applications. Companies such as Zoho have already accepted the challenge of creating a web-based office product. Zoho is not the only company to offer these services; a number of other products have recently emerged in this space as well.
Since Google has an appetite for information, I doubt they would reject the idea of users storing information on their computers rather than their own. The ability to save your documents on Google's servers should be as simple as using the save function and choosing "online" or "offline" storage. As I posted in the past, Google may be considering offering a similar service which could be stand-alone. The concept is simple; provide a free networked drive (G: drive) which can be accessed from any computer by using a valid Google account. The storage space may be shared with your Gmail which provides gigabytes of free storage. A similar 3rd party project already exists, however it is susceptible to even minor changes in Gmail and may cease to function without warning.
With Google being in the "search" business, the technology is available for them to incorporate their knowledge into a live research tool. When composing a message, your "research assistant" will constantly be monitoring the web looking for information on what you are writing about. For example, if you are writing an article about "Open Office", a handy window or panel will show you relevant articles that relate to open office that you can open and read with the click of your mouse. It would be like that annoying paperclip, only useful and not so annoying.
Sharing information amongst groups of friends or co-workers could also increase productivity. It can be accomplished by allowing us to save documents to the "Google Base" or by creating shared folders on your G: drive (which is still purely speculation). Sharing documents should be as simple as choosing from a list of contacts when you save the document. Those people who you share with could connect to your shared folder and open or edit those documents based on their permissions and have them synched up in real-time. Microsoft just introduced a similar concept with its Windows Live.
Finding good document templates for things like resumes can be difficult (and expensive). A large searchable template repository which can be easily accessed is an interesting idea. When a user creates a "New Document..." they can specify the file name and optionally search the repository for templates before the document is created. Right now in OOo you can choose from templates, but only on your local machine. Google isn't in the business of creating content, but they could allow others to create templates and store them in the Google Base for public use. Even corporate letterheads or standard document templates for internal memo's could be stored privately for organizations... these items would be password protected or shared between users.
Google's language tools are also getting better. An integrated translation service which could generate copies documents in the language of your choice would be a helpful feature for some users also. The translation service as it stands still needs work, as the target language can still be difficult to read in some cases. The future looks promising, though, as Google continues to learn the "meaning" of what is being said in the documents stored on their servers. When the language tools become as accurate as a human translator, this type of integrated translation solution would be a great addition to the suite.
Google has the skills, money and distribution power to help Sun make the most functional, intelligent and used office suite. Knowing Google, I can probably even accurately predict the price of such a product/service... free, as long as you are willing to have targeted ads. Of course, Microsoft is already poised to do similar things with its Office system, but unlike OOo doesn't come for free or have open source roots. And, what will Yahoo do to keep up with the other two. So far, it has aligned itself somewhat with Microsoft on the IM and content, but could make nice widgets for OOo and MS Office.