D.B.
Honestly, she should work with a good marketing or advertising company (or an experienced freelancer - there are plenty) who do this. Branding and advertising are professions that demand expertise. If they are a reputable company that manages big projects, they aren't going to go to their friends for suggestions on a company name! All they will end up with is another problem like that one they have.
My husband has been doing this for years, and I can tell you there is no way that even he could come up with something based on the information you have provided. A good marketing professional will work with your friend (who I gather is the CEO) but also the C-suite members (the other "chiefs" such as CFO, COO, CSO if there is one) to learn more about the company, what their niche is, what they've specialized in up to this point and where they hope to go in the industry in view of anticipated changes.
They should expect to spend some decent money on this. But if they're that big a company with this level of clientele you mention, that's a drop in the bucket. They should have a full-pronged approach, which would tie a new name into a new advertising campaign. It's not just about putting a new name on some stationery!
They have to know who they are and where they are going, and to whom they will be marketing their services. They don't need a pun, something cute or catchy, or anything else someone pulled out of thin air. They need a professional approach. They don't need a big expensive ad agency but they need seasoned professionals. That team should include branding experts and a strong conceptual approach to copywriting and graphics - but the design comes separately and further down the line. She should not be going for some cute logo or a lot of bells and whistles that aren't based on anything.
