Final particles (ね, よ, な, さ) can be thrown in the middle of a sentence in conversation as if fillers.
There are technically two types:
- follows the first word/chunk of the sentence
それがね、本物のマツタケなんです That one, see, is the real matsutake.
もしもね、ゾンビに襲われたら If, I mean if, you're attacked by zombies...
It is used to emphasize and draw hearer's attention. This can be also combined with sentence-initial interjections or conjunctions such as あのね, いやな, だけどさ etc.
- used as many as needed between phrases
I think you should... be a bit more conscious... that you have a responsibility.
You must be talking about... a kind of different... type of game console...
This type is often heard while the speaker is giving a long story (or explanation, persuasion etc.), like stopping at each corner to see if the hearer is following you. I can't think of a suitable English translation for this; close to "you see?" or "okay?" but much more lightweight to be used repeatedly.
The in-sentence final particles have no preference of words it attaches to, so you see it wherever after a noun phrase (i.e. after particles), a verbal complex (i.e. fully conjugated chunk of simple or compound verbs) or a free word. Though less revelant to Japanese learning, it is known that their insertable points denote breaking of 文節, a syntactic word in Japanese.
Back to the original question, I think this な belongs the second usage I described, and it has little grammatical meaning here except wanting to impress her name more to the hearer, if any. The choice of な represents a old-fashioned/rustic character, and it would sound rather blunt in everyday speech today.