(mandatory disclaimer - I am no a lawyer)
As far as I understand the license, the fact that you're using monkey patching is inconsequential. The core issue here is that you're using an LGPLed library, modifying its behavior in some way, and then using it. Whether you're monkey patching it, wrapping it, inheriting from it or whatever probably isn't the point here. The main point probably is how tied to that library your code is. If you're providing a general monkey-patching mechanism (e.g., automatically logging every API call), that could be applied to any library, you're probably free to license it as you will. If it's deeply tied to that library (e.g., you're redistributing the LGPLed library and just slightly tweaking some functionality by monkey patching), chanced are it should be considered as a modification of the existing work.