Olympus Camedia E10 won't power on with AA NI-MH batteries but works fine with AA ALKALINE, what can I do?

Always charge batteries before using it for the first time, or if you have not used it for a long period. 3 months is a very long period.
Always charge sets of batteries (2 or 4) together.

Make sure your rechargeable batteries are all good. One defect in a set is killing for the whole set
Don't use barrettes from different sets or brands together.
I had one brand good NiMH batteries and a very expensive charger from the same brand. The charger destroyed all my rechargeable (more than 60 now). When I first discovered the batteries were dropping out, I stopped using the charger, but it did not prevent all batteries to die in the following months. Every time on dropped out I had to change the set. I tried using up the others by grouping the ones with the same Ri, but not longer then a few months and I had a box full to trow away.
Never use a quick charger, capable of charging them within an hour or faster. Us a charger that indicates the Ri is still low enough, so you can see what batteries are bad.
Check you are using a charger for NiMH batteries.
I would try another set (new) because buying Alkaline is also not cheap.
I used 2700 mAh and later 2600 mAh rechargeable, all dead now. since a few weeks I'm now using 2300 mHh from another brand, costing 1/5 of the "best" I had before, and they are doing great. I won't charge them with the quick charger.

Never use manganese (Zinc-Carbon) batteries in electronic devices.
These batteries are only to be used in flash lights.

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