Call Honda and ask them what the factory fill was for your car.
Call your dealer and ask if they sell synthetic oil. Maybe he doesn't and that is why he doesn't recommend it.
From the Amsoil site:
2006 HONDA CIVIC 2.0L 4-cyl Engine Code K20Z3
Engine Oil
Grade 1......API*
Series 2000 Synthetic 0W-30 Motor Oil
SAE 5W-30 Synthetic Motor Oil
SAE 5W-30 XL Synthetic Motor Oil
All Temps......5W-30
We switched to Amsoil 5W30 with the EaO13 oil filter in my son's 2006 Si at the first oil change when the OLM zeroed out at 6,000 miles. I'm not sure he has noticed any difference in the performance of his car. I think he has 10,000 total miles now. We plan to change his oil and filter only once a year, up to 15,000 miles as his driving is considered severe service with lots of short trips to school, church and girlfriend.
If you notice poor shifting in your car like my son did in his, change out the MTF to:
Transmission,.....GLS[1]
AMSOIL Synthetic Synchromesh Transmission Fluid
1. Honda MT fluid, P/N 08798-9016
He (and the Honda mechanic who originally thought the synchronizers were bad and need replacement)notice MUCH better shifting and the grinding into second and the bumping out of third gear was eliminated. I think we changed that out at around 6,000 miles and we plan to change it 50,000 miles or longer if the OLM doesn't come on by then.