Do not talk about working with the client on their own SEO until you've fixed this problem. This is something you caused. Instead, tweak your own SEO so that a search for the client's website doesn't turn up your site. You don't have to remove the website from your portfolio, but there is something very simple you can do to avoid stealing SEO traffic from your client.
Besides, if people are searching for "example", they aren't looking for a website. Consider whether this could have a negative impact on your own SEO by Google ranking you lower for terms such as "website design" or "website development". Good SEO is about target marketing, not mass marketing.
If your client is example.com and folks search "example" and find your site, which has nothing to do about "example" but is instead about building websites, then something is wrong with your SEO. Folks should only find your site if they are searching for terms that represent business you compete in.
The simple solution to this problem is to use the NoFollow attribute on links on your website that lead to the client's site. This tells search engine bots not to let the content from the linked site affect SEO results.
For example, if your client's website is www.example.com, then here is what the link on your own website would look like:
<a href="http://www.example.com/" rel="nofollow">Link text</a>
As for offering to handle the client's SEO, I wouldn't suggest doing that until after you've fixed this problem. If you were to bring up the subject of tweaking my site's SEO when part of the problem was caused by you, I'd be really hesitant to deal with you in the future. I wouldn't be mean about it, but it would definitely affect our business relationship and my desire to give you more money.
What's more, if I saw a review about such an issue, I would steer clear of doing business with you. This isn't to say you have bad intentions, just that I tend to be extremely conservative in who I do business with; I pay close attention to potential red flags, and something like this would heavily influence my decision.