Fascinating question and a couple of ideas come to mind, though they may not be suitable.
-- How about a bill to control administration of senior health and accommodation? There seem to be too many cowboys in the field. One of my friends was, until a few months ago, living in an aged facility in Des Moines. It was sold, closed down, and now she's camping in her car. She's 71 and winter's coming on.
-- While any bill relating to gun control would butt up against the Second Amendment, would it be possible to propose a bill defining the types of weapon people were permitted to carry? Might it also be possible to stipulate that they should be tested for a stable psychological profile?
-- In 1929 when it was proposed that Spanish be used as a second language in Texas schools, the then-governor, Ma Ferguson, stated that "If English was good enough for our lord Jesus Christ, it's good enough for Texas." Would it be possible to write a bill providing special schools where the children of immigrants might learn in their own language as well as English?
-- And following on from Ma Ferguson's comment, what about a bill requiring that candidates for gubernatorial office should not be accepted unless their IQ reaches double figures?