Like other character's Broccoli's name is from that of the company she represents, in this case Broccoli
her nickname, Puchiko actually comes from another of Broccoli's works called Di Gi Charat, from the Wikipedia Article on Broccoli
Broccoli is well known for the Di Gi Charat franchise and series.
in Di Gi Charat there is a character called Puchiko
Petit Charat (Puchiko, プチ・キャラット), is Dejiko's little half-sister and is a catgirl.
there is also character called Gema. looking at Puchiko and Gema from Di Gi Charat and Broccoli from Neptunia they look similar
As you can see both Puchiko and Broccoli look very similar both having the same hairstyle, hat and being catgirls. also Broccoli's weapon Gema (seen under her) looks the exact same as Gema from Di Gi Charat
Also Puchiko and Broccoli both share the same Japanese Seiyuu and English Voice Actress.
Broccoli (ブロッコリー Burokkorī)
Voiced by: Miyuki Sawashiro (Japanese); Karen Strassman (English)
A little girl first appearing in Victory. In Re;Birth 1, she can join Neptune's party near the end of the game. She represents Broccoli.
Source: List of Hyperdimension Neptunia characters - Supporting characters
Petit Charat (プチ・キャラット Puchi Kyaratto) / Puchiko (ぷちこ Puchiko)
Voiced by: Miyuki Sawashiro (1999–2008; Japanese and English in Leave it to Piyoko!), Nao Minakami (2008), Karen Strassman (English)
At the beginning of the story it is explained that Puchiko, also known as Capuccino, is Dejiko's five-year-old sidekick and is a catgirl.
Source: Di Gi Charat - Characters
So Neptune's nickname for Broccoli, Puchiko, is actually a throw back to a character in Di Gi Charat which was made by the company Broccoli