… suggested that a client purchase a domain name.
The name I suggested was her own.
When you own your own name (www.michaelkatz.com, for example), it’s a nice shortcut for telling people how to find you (especially if your “official” web site URL is not so easy to remember).
Depending on how common your name is, it may no longer be available (note to Joe Smith, don’t bother looking). But, yours may still be out there.
Check availability here: www.betterwhois.com
P.S. You can point as many domains to the same web site as you like, so there’s no conflict in having your own name and your official name point to the same place, as mine does.