If you insist on storing data labelled as first name and last name then the best plan is to ask the person for those data items as separate fields (ie. change your web form).

As a marketer first and a programmer second and a DBA third, the form will not be changing! The reason is simple - more people fill in the form when we ask for the name as a single field than if we ask for it in two or three parts.

We've tested pretty much every combination and found (for our audience) that the optimal form uses placeholders, not labels. It has three fields on the initial page which are labelled

Your Name
Your Phone (optional)
Your Best Email
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If we need extra information (we rarely do) then we ask for it after we have captured the key details and stored them safely.

As for the database storage...
We store name information as follows:

prefix
firstname
middle name(s)
nickname
lastname
preferred firstname
preferred fullname
suffix
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The problem I see with storing the full name as they type it into a web form into the database is that we need to use both their full name (in an email to field perhaps or on the front of an envelope) but just their firstname in the salutaion. So we have to split it somewhere and, to my mind, it's best to do this early in the process - i.e. before storing it.

We have discussed adding a separator component for terms like von but we have so few in our databse that we currently treat them as part of the lastname