I'm going to make a bit of an assumption here, so forgive me if I'm off base. It sounds like you are asking about what graphic designers call a symbol mark logo, as opposed to a word mark logo. In that case, we're talking about a BRANDING question and not a UX question. Still, it's worth answering, since it's a challenge that often comes up as part of the overall user experience.
The idea behind a symbol mark is that, for many brands, a graphic symbol can increase unaided brand recognition. If terms like symbol mark, word mark, and aided vs. unaided brand recognition aren't familiar to you, then you've got some Googling to do before pitching the client on updating them.
The good news is that, like most things related to branding and UX, the impact of your client's name vs symbol mark can be tested. In the interest of time, I'll refer you to Survey Monkey's page on setting up a brand recognition survey.
The gist of it is that you need to reach out to your target market and discover whether they recognize the client's brand significantly better with or without the symbol mark. Additionally, you'll want to ask questions about what someone who doesn't know what the company does assumes it means as a symbol, vs a symbol plus name, vs just the name.
At this point, if you believe that users are being confused by a symbol mark, you should pitch your client on a bit of user testing in order to discover more about it's effectiveness. Consider that opening up a conversation by criticizing the design is a great way to alienate the client and impair your working relationship. Tread carefully, until you have quantifiable results!