I am currently reading through this paper and read some of the wikipedia pages on weak measurement and quantum tomography and I am curious if weak measurement could be used to break BB84 quantum key distribution. In particular: couldn't an eavesdropper estimate the states going through the channel? Especially since there are only 4 possible states to estimate.
No, weak measurement and quantum tomography don't break BB84.
I recommend that you create an explicit quantum circuit that implements the weak measurement or the quantum tomography, and check for yourself that it actually fails. The basic problem comes down to the fact that there is a trade-off between how much information you get and how likely you are to get caught, and none of the trade-offs that are available are good enough.
Tomography generally speaking uses a collection of measurements to reproduce an underlying state. So you experimentally reproduce the same situation over and over, collect statistics and find the most likely estimate for that state.
In QKD, information is sent once and doesn't repeat. So for each unit of information, you're never ever able to collect enough data to infer the underlying state.
I should add that Alice and Bob need to make sure that their system overall does not have losses above a certain level, because QKD only works in this circumstance. Too much loss between Alice and Bob means that it might have gone to Eve, and if Eve has too much of the state she can get information out of it.