I must be missing something here. Aosta to Martigny is Grand St Bernard.
There aren't really any towns or villages between the two, though there are some hotels (including
one at the top of the pass - though it's closed at the moment and open only in summer).
If you don't want to stop at on the pass, the logical thing to do is to stop in Aosta or Martigny. Aosta's a whisker less expensive (Italian v Swiss prices), has a lot of Roman bits to look at and a nice main square for evening eating/perambulation; Martigny's like a lot of modernised Swiss towns - clean to the point of being a bit characterless.
The alternative is one of the nearby villages, I guess. On the Italian side, I've stayed in La Thuille on bike trips (on the Petit St Bernard, about 20 mins from Aosta) and it's a charming enough little place. On the Swiss end, the obvious choice is Verbier - though I've only ever been there for the skiing, so I don't know how much of it's open in summer.
Anywhere else – and your choices are pretty much Chamonix-and-its-environs – strikes me as being a fairly pointless detour. Though not as pointless as riding past Martigny to get to Brig, another 50 miles to the east.
If the question was actually: what's a good stop between Aosta and Brig, the answer is... Martigny. (Though Veysonnaz is good: nice climb up the valley sides, epic views - including of light aircraft landing below you at Sion airport down in the valley)