Due to overall cartridge length, it would be very tough to squeeze a .30-06 into a Mexican Mauser 98 action (either the Model 1910 or Model 1934), which is actually an "Intermediate" length with a small ring and small barrel shank. For anything up to an 8x57, the Mexican would have the lightest "starting weight" of any standard Mauser 98 before additional lightening measures. Otherwise, the G33/40, Brno 21H/22F, or a Husqvarna HVA action are all standard length actions with small ring and small shank, and would be next lightest starting point. All would be suitable for a .30-06, and have been frequently built in that caliber. In addition, most of the Husqvarnas came from the factory with an aluminum triggerguard with hinged floorplate, which saves a few extra ounces, and the design also has a very tidy bolt release. The G33/40 on the other hand, has additional lightening cuts on the sides of the action, and even came with a hollow bolt knob for extra weight savings (although it is frequently replaced during customization). However, the commercial Husqvarna HVA did not come with a stripper clip slots, which leaves it out of consideration if that is a factor (rapid reloads for charging herds of... something?)

However, once you have a decently light action (any of the above will do), most of the potential "practical" weight savings will come from the barrel profile, stock design, and scope set-up. Even if you choose a wood or laminate stock (not synthetic) with an appropriately slender profile, additional weight can saved by hollowing excess wood under the buttplate, and inletting additional hollows inside the barrel channel (like some military stocks). Also keep in mind that not all walnut is the same density; Claro is generally less dense than English, and a plain grade stock with straight grain is generally less dense than one with a lot of burl or figure. Again, we're only talking a few ounces, but those ounces add up.

Another place to save weight is in the choice of scope bases, rings, and the scope itself. It always amazes me to see a custom lightweight rifle or carbine burdened down with a huge variable power scope, clamped in a set of chunky steel rings, sitting on an equally massive set of bases (or worse yet, a one-piece steel base). This is another area where a little homework can save a lot of weight, for little or no extra cost.

As for the Springfield 1903 action, although it is technically a small ring design, in issue form I believe it is comparable to a large ring Mauser 98 in weight. The 1903 receiver has some extra protrusions in various places, as well as a hefty upper tang and chunky cocking piece that add unnecessary weight. Although it is a natural choice for building a .30-06, it wouldn't be my first choice for a lightweight custom.

Although often overlooked, the Swedish Mauser 96 action also makes a fine lightweight rifle, especially if you stick with traditional cartridges (6.5x55, 7x57, 8x57, etc) and stay away from high-intensity cartridges (.308 Win, short magnums, etc). However, the Swede has a cock on closing design (instead of cock on opening, like Mauser 98s) which some people do not like in a sporting rifle. Although it can be converted to cock on opening, it is just another thing to deal with during the conversion. Personally, I would look for a later commercial Husqvarna version of the Model 96 (known as the Model 46) if I were going to build one, since it still retains the stripper clip slot on top, but does not have the thumb slot on the left side of the receiver found on the military rifles or pre-war sporters.

Like John303, I also have some historic sporting rifles in 8x57, built on the WW1 era Kar98 action (aka 98a). It was a standard length, small ring action with a large barrel shank, and would be another option to consider. Although dimensionally identical to the G33/40, it is generally not considered as strong as the G33/40, since the receiver walls around the barrel on the Kar98 are not as thick. The Kar98 / 98a is sometimes referred to as the Erfurt or Danzig carbine, after the two arsenals where they were made. Although probably not the first choice, they may be more common in your part of the world and therefore more affordable, and would also make a fine lightweight in a traditional cartridge.