I have worked with a client with a similar issue. Their competitor was bidding on the brand name of my client. You will have to donate some Adwords spent and create “branded campaign” where you focus on bidding on your client’s business name. Your client should have a high quality score for their brand keywords since their website should but an authority on the brand name.
As far as stopping a competitor from bidding on the name the most you can do is send a cease and desist letter and I would only recommend this if the brand name has a registered trademark.
As others have mentioned, if the competitor is bidding on your client's brand name then it may fall under Google's trademark policy and you can file a complaint with Google.
If they're bidding on generic keywords however then the only thing to do is ensure that your quality score is as high as possible, which will reduce your ad cost and push you above your competitor whilst costing you less per click.
Most of the copycats that I've come across aren't very good when it comes to campaign optimization, meaning that most of the times it's fairly easy to achieve a much higher quality score than they'll ever see.
Make sure that your ad copy matches your keywords, that your ad copy + keywords match the content on your site, that you're linking people to pages that have good content (as opposed to "naked" landing pages), and you'll outrank your competitors shortly.
maybe something not yet mentioned is to actually protect your brand by registering it (through your national registering agency, OHIM or WIPO, as examples).
You should know if your brand should be registered by calculating the profit driven by brand traffic vs registration costs.
Sure, advertisers will always be able to bid on your brand, but as you register it (first) and let Google know you want it to be restricted (second) you will succeed in having many advertisers using your brand in their ads, to have their ads disapproved.
They might then change the ads, but won't be able to mention your brand in their text ad anymore, hence become a worst Quality Score. With a worst QS it will be less profitable for them to keep bidding on your brand intensively.
I have successfully implemented brand defence strategies for my customers, increasing profitability on both paid and organic channels.