I wouldn't worry too much about ECDSA being broken. If that happens in a reliable and fast way bitcoin is probably the least of your worries.
You could always introduce a protocol update that supports even stronger encryption aswell. All you'd need to do is transfer the coins from old addresses to new ones.
If all cryptography becomes useless well then goodbye freedom, hello Orwellian state.
You introduce a new address format accepted by the protocol next to the old one. Anyone wishing to obtain the stronger encryption sends their own bitcoin to such a new address
Those that don't move their coins risk being attacked. We could then argue what will happen to those coins that are now lost but then become available through brute forcing.
In a sense it would be like mining bitcoins.
[edit] think of how you are now using change addresses. all that really is required is that the protocol accepts such a stronger address as a valid output