Sometimes a great deal -- as when the name is "hot" and will materially aid the public in conceptualizing your business. Sometimes nothing -- as in the case of what is intended to be a local business, such as a local dry cleaner. Its prospects are not likely to be influenced by its technical corporate name. If you expect to be a major player, a name will be an important asset.
Picking a corporate name is not as simple as it sounds. Others may have a prior claim: (1) it may be in use or be someone else's federal or state trademark or service mark; (2) it may be a generic name so that someone else can readily copy it; (3) it may be used by (or similar to) another existing corporation. The acceptance of your corporate name by your state corporate office does not guarantee your right to use it if others have trademarked the name.
And once you have found a catchy name, move fast! File with the corporate office on an expedited basis, particularly if you feel that the name has "magic" to it, and start the process to trademark it. Your competitors might be seeking approval of the same or similar name and are moving faster by filing incorporation or trademark papers "a step ahead" of yours.