How important is your search fund name to investors?

I started a company in business school which was simply my last name- tax paper work filed and all, now that I'm gearing up to prep for a funded search, I am curious how much weight a fund's name carries vs. connections, networking, etc. for funders. Any thoughts?

As an aside, it seems like all the cool fund names have been taken! Does it have to be 100% unique to avoid potential lawsuits from all the other 'XYZ Capital' firms out there?

Thanks!