Is my attic floor over my garage strong enough to use it for storage and maybe as a small work area?
You have 2x6 joists 16" o.c. and lets assume #2 fir- the MAXIMUM spam (where they are supported underneath - bearing the weight) for a 20 lb dead load is 9'10" and for 10 lb dead load: 13' 2"
You show 11' +- in photo but it is unclear if there is support at that point.
you can check span table at:
https://www.awc.org/pdf/codes-standards/publications/wfcm/AWC-WFCM2018-ViewOnly-1711.pdf

There is some interesting framing in that attic: The 2x4 at the peak of the triangle which is set on top of the ceiling joists is very odd, in my opinion.  It is transferring a load from the roof to the ceiling, and unless there is a wall underneath: you probobly have a bow in the ceiling at that point below.

Soooo: minimal storage, depending upon support at the 11' 3/4" point - Once you put a floor downs there even with the intentions of keeping it light up there - people forget in time.. . . . 

As to the beefing up the joists with 2x8s that is an option - but the total span is more important. Is the entire width unsupported? and if so what is that total span?