Here are some obvious choices:
There is the 17-55 f/2.8, but this is not intended for full frame (FX) and is large and heavy, being the "pro" lens for DX in this range. It's appropriately sized for the D300, however. No image stabilization. I personally did not like the optical quality of this lens (two samples were examined) because it was actually of poorer quality than my 18-70 under many conditions (though better under others).
For a lighter "normal zoom" lens, there is the wonderful little 18-70 f/3.3-4.5. Also not for FX and no image stabilization. You can consider it the "f/4" lens in this range. It's inexpensive but is a definite cut above the typical consumer-level lenses. Image quality is truly excellent except for some vignetting wide open at the ends of the zoom, and some barrel distortion at 18mm. It's a remarkably light, compact, versatile and high-quality lens that is fit for a D300.
There is the 18-135 (DX, no image stabilization), but that is mostly a f/5.6 lens that is very sharp but has some poorer construction and some optical issues such as chromatic aberration, and which I do not consider worthy of the D300.
There is the 18-200 (DX, with image stabilization) that a lot of people like very much. It's also basically an f/5.6 lens through most of the range. Construction is similar to the 18-70, but with a much longer extension that feels a bit wobbly to me. Image quality is very good over-all except for rather bad distortion at the widest angles.
So, for a lighter "f/4" normal zoom, I prefer the 18-70. I mostly use it on a D40X, where the combination is so light and handy and produces such high quality that I actually tend to prefer it over my D200 plus 17-35 f/2.8, etc.