OK, I'll give that a stab.
From the Li-Ion cell
Wikipedia entry, lithium can theoretically produce 11.6kWh/kg. Your 3.7V 3000mAh cell contains 11.1Wh, which I'm going to approximate to 1g of lithium. Battery University
agrees - their approximation of Ah x 0.3 gives 0.9g.
Of course, cells aren't perfectly material-efficient - far from it - so I'm going to adopt your 2g of lithium as a reasonable-sounding guess at what's in a real cell.
Now, I'm not going to use volumes or densities here, because the lithium in a
Li-Ion cell isn't metallic, it's part of a chemical compound, whereas the aluminium in a charged
Al-Ion cell is metallic. Instead, I'll use the
mole, which is a standard chemical unit containing
6.022×1023 atoms of an element.
Lithium has a standard atomic weight of 6.94, so 2g of lithium is 2g / 6.94 or 0.288 moles of lithium.
Aluminium has a standard atomic weight of 26.98, so 0.288 moles of aluminium would weigh 0.288 x 26.98 or 7.77g.
So far, so good - an Al-Ion cell containing the same "amount" of aluminium will weigh 5.77g more than the Li-Ion cell.
However, aluminium can contribute three electrons per atom to a chemical reaction, where lithium only contributes one. That means that the Al-Ion cell could have three times the charge capacity for the same "amount" of reactive metal, so this cell would be a 2V 9000mAh cell containing 18Wh of energy.
An energy increase by a factor of 18Wh / 11.1Wh or 1.62 implies 62% more runtime, assuming the same driver circuit efficiency. In relation to that, it's interesting to note that two safer Al-Ion cells in series would produce 4V - a drop-in replacement for any driver intended to use a single Li-Ion cell, although there'd be a penalty for the extra structural material in two cells. You'd basically have to decide whether it was better to take the efficiency hit in the driver circuit or the extra cell materials.
If the cell volume is similar, then the extra energy can offset the weight penalty. However, that's a big assumption and not one I know enough to test. Among other things, it assumes that you can get all those extra electrons out of the aluminium without having to add any other extra reactants (more weight and volume) to do it.
Having said that, scaling your 44g 11.1Wh Li-Ion cell up by a factor of 1.62 to 18Wh gives 1.62 x 44g or 71.3g, where the Al-Ion cell would weigh 44g + 5.77g or 49.8g, so the equivalent-energy weight penalty would be -21.5g. In other words, Li-Ion cells with the same energy would weigh 21.5g more than the Al-Ion equivalent. They'd also be a factor of 1.17 (cube root 1.62) larger in linear dimensions. That's quite a lot of leeway in weight and volume for getting extra reactants into the Al-Ion cell.
Conclusion: I'd call 5.77g negligible, and in fact, for a safer, faster-charging cell with more energy in the same volume, I'd delightedly accept those few grams of extra weight. Extra reactants would reduce the theoretical 62% extra energy quite significantly, but there's definitely potential in this new Al-Ion cell idea.
