i had this exact same problem in my 95 accord also.

a few seconds after u start up the car, the ABS motor makes a little grinding noise (may take up to a minute to that).

doesn't do it every time, but just randomly. about 6/10 times.

anybody here has had that problem before?
any solution w/o having to replace the whole motor?

 

I pretty sure it's the ABS module engaging. Mine does it too.

 

Mine used to do that every once in a while after a cold start. Never lasted long, would just cycle on for a few seconds when I started to drive and then stop. Figured it was just going thru its startup process. I wouldn't worry about it.

 

It's normal.

 

Normal. Have no fear.

 

but did they do this when the cars was brand new too?

 

Yes, guy at the dealer told me that is was the ABS engaging. That's why it only occurs at start up.