In a thunderstorm I was thinking the following: suppose I am rowing in a lake during a thunderstorm. How big a Faraday cage do I need to make to protect myself? If lightning strikes the cage, will it dissipate away through the water?
The boat itself must be enclosed within the Faraday cage. Or, alternatively, let's assume that your boat is your Faraday cage, i.e. your boat is a floating enclosure made of conductive materials.
In theory, you can make such enclosed boat. Or such cage that contains your boat. Yes, the lightning current will be dissipated in the waters (the lightning also loses energy by heating the water before reaching earth). Make sure the Faraday cage is made of perfect conductors.
By using perfect conductors, there will be no electric potential gradient in your boat, although the gazillion-ampere current is flowing there. Ohm's law says $V=IR$, and obviously $V$ is zero when $R$ is zero (as in perfect conductors). You are protected by your Faraday enclosure.
The lightning current, seeking least resistance, will travel through the cage rather than your body.
However, in this situation, there is a possibility for real-world conductors to introduce electrocuting voltages across your body. Their $R$ is not really zero.
Well, actually, there's a solution. Lining the inner side of the cage with good insulator provides protection to voltage gradient in your boat. Cars exterior are mostly conductive (that makes good Faraday cages) and they are lined with insulation inside, that's why it is safe to hide in a car during a bad thunderstorm.
What may be required in this situation is not really a Faraday cage but rather what is known as a lightning conductor which is a system of conductors designed to provide an alternate path for the lightning to flow. You mentioned that you are rowing which indicates a small boat size and probably one which does not have sails or a mast. Materials such as wood which boats are made mostly of have a lower dielectric breakdown strength compared to the surrounding air. So they are likely to breakdown first providing lightning a path to reach the ground. If conditions are conducive for lightning to strike in your vicinity, then any kind of protrusion above the surface of water, such as the boat, or even a human being seated on the boat can be the first target for the lightning to strike. The point of a lightning conductor is to offer itself as a target instead of the human occupant or other sensitive components of the boat. Please note that while the lightning conductor may divert the strike away from the human occupant, it does not prevent the strike itself. In fact there is the possibility that it may actually cause the lightning strike by acting as a 'magnet' for the lightning due to its good conductivity. Hence, the entire system needs to be designed carefully. Even a well designed system cannot guarantee safety. My best advice would be not to venture out in thunderstorms and take note of the weather forecast before venturing out.
Of cours one need a Faraday cage on that boat. Just a "lightning rod" would be equivalent to a stroke passing You at the distance to the rod. The field in the vicinity can kill You!. On a boat, You need a cage and the cage should have some straps of metal extending into the water under the boat. How big? If its a closed metal housing, it could just cover you like armour. If its some "mesh" the distance of that mesh to any part of Your body should be about the diameter size of the mesh holes. And: the mesh has to be able to carry some thousand Amperes for a millisecond or so.