How can I approach a non-technical boss about contributing to open source in a way that is likely to yield a positive result?
What non-technical people hear when discussing contributing to open source is their time and money being given away free. They see your time and energy spent on promoting the open source product and dealing with the community instead of doing what you are paid for.
You need to focus on the cost/benefit of open source, specifically how they will make more money. You also need to dispel their fears about your attention and time.
What arguing points would be relevant to a non-technical boss when speaking in favor of contributing to open source? What arguing points should I avoid?
First, is this tool a competitive advantage? If your company's main business is around databases, testing or software development, the code will not be open sourced. If quality or release cycles are your competitive advantage, the answer will also be no.
Second, is it something others would pay for, given the additional time and resources for productization? In other words, if they can sell this now or in the future, the answer is also no. Many non-technical people would rather have the option to sell it in the future, too, making this argument difficult. They also may fear others selling it, so a restrictive license (e.g. GPL) may be worth mentioning.
Assuming neither of these questions poses a problem, think of the benefits contributing to open source would bring. For example, others could fix bugs or add features (examples of similar software would be useful here). It could spread the company name in technical circles, making hiring new developers easier (examples from other similar companies would be useful). It could increase morale inside the development team (get others' agreement on this).
Your management need assurance that you will focus on your main role and not the open source product. Perhaps agree on time commitments (e.g. 1 hour per week), how it will be distributed (e.g. if it is on github, who owns and pays for the account?) and what license it will use (e.g. MIT/BSD to limit liability and encourage use or GPL to ensure it remains open source?).
Contributing to open source is really a cultural question for an organization. This is much harder to quantify but developers are often motivated by respect of their peers as much (if not more so) then salary. If you use lots of open source it can be nice to give back, too. Unfortunately, this will not hold sway with more traditional management.