I will go with fuel or oil level.. Also the brushes could be worn.
The Onan generators are smart.. If anything prevents the generator from generating power it will shut down.
Check your button flash code for an idea of why the GENERATOR thinks it shut down One code is "Cranking speed too low" This can be due to failure of the generator to generate.. The one time I have seen this the owner had to replace the brushes.. (NOTE: I don't fix 'em, I just remember what it requires to fix 'em when others get 'em fixed).
Fuel filter, Air filter and Oil level are all good ideas too. IF it runs at all after you release the button on a cold start these would be my first choices.