Can you be more specific about why the code is bad? There are a limited set of architectural reasons for which I would support a complete rewrite. They typically correspond to non-functional requirements: 1) scale - the system architecture simply cannot support scaling 2) reliability - the system architecture is built in such a way that the system often fails 3) multi language - the system needs to be modified to support multiple languages 4) security - the system needs to be modified to support a more robust security access model etc (there are many items like this)
These things all touch every single bit of code and so could warrant a rewrite. Im skeptical of most other reasons. In most cases you can just modularize the most dangerous aspects and you will be fine.
Likewise, function names are ideal to refactor -- many IDEs will automate this for you.
I don't know your situation, but this sounds to me more like framework envy (and I base this on experience... I've worked with projects on a lot of crappy and/or homegrown frameworks before). If you haven't actually done any major projects in the newer framework yet, that's a bad sign as well, I'm afraid. :)
On a more positive note -- it might be possible to incrementally move to a new framework. It's harder if you want to switch languages/environments completely, but if not, it's probably quite feasible (switch over a feature at a time). If both frameworks can access the same session storage... that's likely the main tricky aspect.
Then I would provide the benefits of doing both and then I'd present this to the person and say having, looked at our current code I believe that we should rewrite it because of X, Y and Z, in fact I'd even summarize this at the bottom of the table as well incase they want to take the table away and think about it.