Some wealthy benefactor sends you a letter stating that he/she will finance you in any research & development project you desire. Provided it’s frivolous. The benefactor already supports many charitable organizations and has a few million lying around to help the average Jo(e) fulfill his/her wildest dream. No idea is too crazy.
[Karl Malden]What will you do? What WILL you do?[KM]
I’ve always wanted to devise a way to record dreams so that I can watch them the next day.
To see how long it takes me to get sick of traveling first class all over the globe… 
Flying robot monkey butlers!
or 1920’s Style Death Rays
Seriously, I’d love to invent a floating home. When my wife and I were in Greece last year we saw a prototype home that was tethered to the bottom of a small bay. It was 2200 sqft and had all the amenities of home…
My home however, would use solar power almost exclusively and would have all the cool hightech stuff needed to reside on the Ocean…Including an underwater garage where my on-road / submarine car would go, and the ability to submerge during storms. Scrubbers for our air incase we needed extended stays under water…
Self cleaning houses and dry cleaning closets.
Doubt that would be frivolous.
I would like to research why elederly, retired people move to Florida. I’m sure that I could make a PhD dissertation out of this one, and also appear in front of congress.
The artificial appendix.
Easy! Design and build the ultimate Formula One car! Gonna need more that “a few million” however to best the likes of Ferrari. Williams and McLaren. I would attempt to build a car so superior that even I could whip the field! I’d score my first World Championship, then retire.
Then, my kid takes over, and systematicly shatters every record imaginable.
X-ray spectacles which don’t penetrate skin. Think about it… regular x-ray vision would really be pretty useless unless you’re an orthopedic surgeon…
World Eater’s “old tyme” Bacon jerky.
MMMmmmmmmmmmm 
Train marmots to throw lumber, and measure the quantity the trainees can throw.
An automated rooftop snow removal system inexpensive enough to put on the typical house. Man, do I get tired of climbing up there and shoveling the damned thing off!
By the way…this is probably one reason why older people tend to move to warmer climates, like Florida.
I want a serious biomedical investigation, with MRIs, biopsies, and even autopsies as needed, on why knuckles make that funny sound when you crack them.
Research into how people could eat whatever they want and maintain normal body weight. Maybe we could take it in pill form.
Because I love food and don’t want to diet.
Cigarettes that don’t cause cancer and don’t taste like ass, so I can smoke all I want in public and not get glared at. Oh, and don’t leave stinky smoke on my clothes.
Is that frivolous enough?
If not, heads-up display maps in cars. I hate looking at maps while driving.
OR
A permanant odd-colored (blue, pink, purple, etc) hair dye. That should be frivolous enough.
A real dietary supplement that increases penis size. That would be a gold mine.
Ditto for a pill that actually does increases breast size.
I’d fund a study that would find a way to make egg nog have no calories and taste exactly the same.
A fluid-bath-bed of some sort in which you would sleep fully submerged in water or some other fluid, but could breathe (through a mask, or whatever). Being enveloped in water is SO COMFORTABLE.
A miniature, flying, talking dragon made of chocolate that I can communicate with telepathically. It spews cappucino with my choice of vanilla or caramel flavoring on command. It also doubles as a remote for the TV/DVD/VCR/stereo.
And his name needs to be Frank. This is important.
Oh, wait. You said frivolous.
Ummm, how about chameleon skin (you know, like whats-her-name, Stamos in the XXX-Men movie)?