I guess the aerogel would be dense enough compared to hydrogen that a conventional balloon would be better in all respects though.
Still curious as to whether it's been done already though.
Or lightweight solar panels, I suppose, feeding an air-source heat pump to boil the water. Normal Li-ion batteries sadly don't really have enough energy density (water takes 620 Wh/kg to boil; commercial energy-dense Li-ion is in the 200~300 Wh/kg range; at sea level and boiling point (=lowest possible temperature without it wanting to condense on every dust particle and other condensation nucleus) water vapor is half as dense as 20°C air so it could lift itself twice (and once it's used to lift itself already), and even a very good heat pump will have severe difficulty reaching a COP of 4.0). So I guess if you try really hard you could make a self-inflating steam balloon that takes off using Li-ion fed heat pumps and later stays airborne using solar cells to continue feeding the heat pumps...
Also - "water is lighter than air" is an odd claim.
But no, water is lighter than air. In it's gaseous form. Which requires substantial temperature or low pressure. The former you can make easily, the latter you encounter if you fly high enough.
- Helium and hydrogen “escape from everywhere”.
- Hydrogen burns
- Helium is obtained from natural gas and is rather expensive.
An aerogel made from biomaterials (think of a sort of bio-engineered coral) could possibly have enough strength and low density.
Edit: Wikipedia has a persuasive picture of some aerogel supporting a house brick!