No rule says you have to finger those top two notes, b&d, in root position. Any two notes in the chord, anywhere on the neck would do fine. For example, d (second string, fifth fret) and b (first string, seventh fret) would work, since a G chord is g/b/d.
Swap back and forth between these two a couple of times and you'll hear that they're different voicings of the same chord.
The shape's slightly different -- the fingers are separated by a fret instead of being adjacent -- but you're only working on two strings, so it doesn't take long to get comfy switching from one to the other.
Personally, I use my index and ring fingers for the second shape, but your mileage may vary.
I'm going to want a shorthand for these two shapes, so I'll name them "near" and "far." In the root position, the fingers are near one another (actually, right next to one another, probably touching). In the one up the neck, they're farther apart, with an un-fingered fret between them.
Any time you're playing a bunch of G chords and get bored, just slide on up the neck and play far instead of near. The bass rings while the treble moves.
(This idea transfers cleanly to D chords on the guitar, whose bass use another pair of open strings. I've watched Patti Cummings use "slud-up" D chords a lot.)
You're nodding, "Okay. Easy enough."
It gets better, but not harder.
Next, instead of taking the express, take the local. Start down in first position with a normal G. Next, slide up just one note for each finger: b->c, g->a. Skip the open bass on this for a second and just do the treble. Fiddlers will tell you you're fingering these in second position.
This is a far, but it's not a G chord. (It's a D7, or at least two notes of a D7).
Finally, slide that on up to third position, to the far G you were playing earlier. Two shapes -- near and far -- and three positions -- 1, 2, and 3.
We're there. Let's do something with those three. Play 1, 1, 2, 3. Sing "Yan-kee Doo-dle." You're playing the melody with double stops.
Let's add a fourth double stop -- a near with your ring finger all the way up on the eighth fret. Old shape, new position. Call that 4. Play 4, 3, 2, 1. Sing "In his cap and ..."
If you didn't immediately leap to 1, 1, 2, 3, 4, 3, 2, 1 "Stuck a feather in his cap and ...," shame on you.
Plectrum banjo players, like Harry Reser and Eddy Peabody, take this idea and run with it, playing chord melodies. Every note gets replaced by a chord, topped by the original melody note.
I've never seen anyone try chord melodies on mandolin. (You could be the first!) I'll bet that's because you'd have to get good and practice and stuff. If I were going to go to all that work, I'd probably just go whole hog and take up a real musical instrument, like fiddle or plectrum banjo or kaval.
However, even on a mandolin, tossing a double-stop or two into your melody from time to time, or even a run of double-stops, adds color without a lot of work. Whenever the melody goes up on the high, e string, hunt for a near or a far shape that sounds okay with one of the melody notes on top, and use that double-stop instead of the single note.
