Definitely west side if you are planning on much in the way of diving. The Big Island is just that (Big) and for the most part staying on one side of the island and planning day activities on the other is impractical. A trip to the opposite side is worth a long day or two, but otherwise I'd pick a side and more or less try to stick with it. Many first tiem visitors get misled somewhat by looking at mileage looking at the map, not realizing our 60-80 mile long west side "highway" is a 2 lane road that's been over capacity for the better part of a decade. Add the construction that is now occuring in several spots and driving can take longer than expected. If you stay somewhere in the stretch from Puako to Captain Cook you should be able to see lots, and get in your diving, while only driving a few minutes to an hour or so. If you stay all the way down at south point, or nort up at Honokaa, you'd likely be an hour or two away from anything nearly every day.
The west side is more touristy, generally sunnier and much better diving. The east side is more "old Hawaii", generally wetter (I think Hilo is regarded as the US's wettest or second wettest town) and gets the brunt of the surf generated by tradewinds. Hilo side does have the volcano, which is a huge draw, and also more of the rain forest type vegetation, while the west side can vary between lava desert and tropical vegetation depending on where you are staying.
Have fun,