It doesn't necessarily matter if the solar panel can charge the battery fully in a day. It's still better to have more capacity and might even keep the battery from overcharging, depending on the size of the solar panel. "Probably" not, the solar panel is probably small and cost economized, but in a good design it would be large enough to charge the battery on shorter, cloudy winter days which is a significantly lower amount than on a long, sunny summer day.
Anyway, there is a downside to the higher capacity cells, all else as equal as it can be. They would have fewer recharge cycles, probably are NiMH instead of NiCd, which may also mean lower performance in low temperatures too, but if they have fewer recharge cycles it is a better thing to not have them drain by as high a % every time.
A cycle means a full cycle so if a smaller % is taken out and put back, it counts closer to that percentage of a cycle than to 1.0 cycles daily. Granted, 1 x AA cell is not expensive and the solar panel on the light will probably degrade from UV within a few years anyway if it is the typical cheap designs I've seen that use a 600mAh NiCd cell, where the solar panel is either epoxy encapsulated or behind a plastic window panel, either of which get hazy and progressively block more and more light.
The other variable is that not all 2400mAh NiMH cells are created equal. That may or may not be an accurate rating (could be too high as is often the case with less than major brand cells) and they may or may not even exceed 2400mAh in such a low drain application.
Personally, I would keep using the 600mAh cells until their runtime is insufficient, then switch to a quality ~2100mAh LSD cell like Eneloop.