There are a couple of different issues with the Gen I Busas.
Up to 2003, there was an issue with the starter clutch, as Golden Child mentioned above. Easy to replace. Symptom is when you attempt to start the bike, the starter grinds a little and then stops. If you stop pressing the starter button, and then attempt to start again, it frequently will fire right up. It's almost as if the starter bogs down at first, then when you get off of it and try again, it will work. There were times when I was able to bump the bike against the transmission just a little and it would fire right up. Frequently happened at gas pumps after fueling. There were times when I thought it was heat related but I was never entirely convinced of that, especially since replacing the starter motor solved the problem.
The other issue IS related to heat. If the bike sits in the sun and heats the fuel in the tank then the bike may become hard to start. I have no idea why except that it has something to do with the build up of pressure in the fuel system. I finally determined that if I open the gas tank and relieve the pressure before starting, it would make the bike much easier to start.
Symptom was that the bike would crank endlessly and may even fire up for a moment before dying. If I held the throttle wide open while cranking, it would eventually start. After cranking endlessly, the bike would start but would run very rough and if you didn't hold the throttle open so it would stay about 2,500 rpm, it would die again. After allowing it to run with the throttle open for about a minute, it would smooth out and run just fine after that.
Aside from opening the gas cap before starting on hot days as mentioned above, I never have figured this one out. Very strange. It still does this, btw.
--Wag--
But then you´d have to upgrade the starter clutch and it´s not very cheap if you can´t find used stuff, then i guess this would be the ideal fix:
Hayabusa Sprag Starter gear. - Hayabusa Race Engine parts Turbo Motorcycle and Performance Bike Tuning by Big CC Racing
I´ll start checking connectors and fuel lines then, sometimes my clock gets reset also, i happened to look at the panel last time it happened, it was like turning off and on the ignition while riding, all meters shut off and bottomed out like when it does the start check but the bike kept running without reacting.
I agree there yes, it seems to be a heat problem, mine acts a bit different though, it ALWAYS start 3 or 4 turns, hot or cold, and it never kicks back so it stops, it´s just hesitating and a clunking sound just before it starts when it´s warm. I´ve read that it´s important to have the fast idler adjusted correctly also, i never got why since when it´s off it´s off and doesn´t open the throttle at all.
The previous owner had ONE kickback a couple of years ago and replaced the starter and says the problem disappeared but it´s here now for sure, here´s the pics he put up on a swedish forum: