I mentioned it in the old thread and it happening again from time to time:
https://www.fitfreak.net/forums/fit-diy-repair-maintenance/51721-solenoid-problem.html
The similar problem was VERY common on 2nd generation Toyota Camry when they were close to be 20 years old.
On my Fit it began much earlier (when it was 3 years old)
While on Camry it is caused by increased resistance of starter solenoid pull-in coil, on Fit it most likely caused by starter itself (to be more specific, the brushes)
The brush holder assembly is not very expensive but taking the starter out is very fun
I am in the process of making the special tool for upper 14 mm starter bold as it seems to be super tight.
Will post updates soon!

 

I'm assuming you've checked the easy obvious stuff first, like that there actually is voltage across the solenoid when it should be cranking and stuff like that.

I had a car once where the starter would cut out occasionally, mostly after changing the oil. it was due to a loose wire on the solenoid, and it was loose because it was necessary to reach across it to get at the (nearly inaccessible) oil filter. The filter location was one of a handful of things on the car that I was sure must have been designed late on a Friday afternoon.

 

make sure the dash lights stay on while trying to crank. If they go dead, you simply have a battery wire loose.

 

On my Camry I just connected the test light to the short thick wire from solenoid to starter motor(light on but no click indicates bad brushes)
On Fit this is a lot harder but probably will be my next step. The lights are staying bright.
Same problem on 07 CRV ended up to be 1 brush worn out (I took apart an old starter)
I am also planning to bypass the starter relay