Find the logos of the bigger, better, grander competitors of theirs.
All of them. Put them into a beautiful PDF that goes through each and every one, and then shows them all laid out together.
Then ask them which one doesn't belong.
If they point to theirs... you're in business.
EDIT: ADDITION: Partially in response to the question posed below in the comment about whether or not this actually works, and partially because this is an absolutely essential part of doing any image/identity creativity on behalf of anyone, ever:.
Not only does this approach work, it's fundamentally important to the process of (re)designing a logo for any brand. In the research stage you must analyse all those companies, brands and products in the same space, above and below, for their imagery, iconography, colouring, themes and attitudes as presented via their imagery, identity, advertising and marketing.
Just by virtue of being a designer you should be keeping up to date on the world's design, in general and specific to your field of interest and endeavour and always be the best conversationalist on the zeitgeists within it.
Like it or not every brand/product/company/service exists in a space within which you're designing a component of their armoury to duke it out for revenue, reputation and reverence.
// You should deliberately position a company within that space by designing with full understanding of the constructs within which the brand will be seen and perceived.
If a company doesn't understand its identity, and you think you do, the fastest path to demonstrating the position of their brand's image/identity is by contrast within their market. Even if they can't see their own flaws they'll recognise the strengths in others.
If you're right, and they're sentient and self aware, they will both recognise their competitors qualities and their own perceptions of them, whilst concurrently realising you (as a designer) are actively assisting with their market positioning via their image and identity. That you GET where they're situated and what they're up against from an image and identity perspective. You just became an ally.
In this way this exercise builds trust AND understanding.
There is an IF to the sentence above... if they don't recognise their own logo as not belonging you're either misguided in your efforts or they're too blind to see/recognise any of their own deficiencies. You can't help people lacking in self awareness, and if you're misguided you're only going to do damage. Find yourself in this spot it's best to just move on.
But if they sit back after reviewing all other brands in their market, and look at you thoughtfully, then you're in for a genuinely meaningful relationship. And things are about to accelerate. So you'd best start listening very carefully to what he says next, and respond even more carefully to any questions he postures.
You could be about to begin transitioning from designer to Designer. Or being shown the door if they think you're only able to identify the problem but unable to provide a solution. Conversationalists listen more than they speak, and ask more questions than they make statements.