Based on my research, "foundation", "Institute" and "center" are the most common nouns that are used in the names of nonprofits in approximately that order. "Center" might be inappropriate because the organization in question will probably not be based out of a single building.
In my view, the most important things are that the name should clearly communicate what the organization does, should not sound cultish, and should sound good in the same sentence as "the singularity Institute". (This may rule out "Institute" as well.)
Here are some names of mine. Many suggest the emphasis of System 2 over System 1. Others suggest improvements to the process of thinking itself, as opposed to being more correct than others about something.
And some clever and probably bad names:
If we take this--"One obvious question: when is the name most important? When first heard; Introductions."--seriously, then the simpler the better.
Hence, more descriptive names with a higher syllable count, like "deliberative thought foundation" or "foundation for improved decision-making" are inferior to names like "the better decisions foundation".
Another consideration: don't pick something obviously pretentious, like "the primate debugging group", nor something less obviously pretentious, like "the...
The Singularity Institute wants to spin off a separate rationality-related organization. (If it's not obvious what this would do, it would e.g. develop things like the rationality katas as material for local meetups, high schools and colleges, bootcamps and seminars, have an annual conference and sessions in different cities and so on and so on.)
We can't think of a name for this organization.
We can't think of any names that seem good enough to be audience-tested.
We don't have any ideas good enough that we'd want to mention them in this post.
Help.