Without much warning, she's asked for a name and gives one which is real, but not strongly linked to Harry. There's a chance of getting away with it, if the Snatchers don't check too carefully.

She has four options:

• Claim to be pure blood - but this is a relatively small group, they know each other, and high risk of being caught out rapidly, if not immediately.

• Claim to be a Muggle - instant target.

• Claim to be Muggle-born - again, instant target.

• Claim to be a half-blood - until Scabior says that's 'easy enough to check', there was a chance that this would be a status low enough to be looked down on and dismissed, without anything more than some verbal abuse along the way. She didn't know for certain that the name she gave could or would be checked to confirm her blood status. In a selection of bad choices, it was the one that gave her the best chance.

Penelope‘s blood status is not known.

Tom Riddle said that Ginny set the basilisk on four Mudbloods and the Squib’s cat. Three of the Mudbloods are for certain Colin Creevey, Justin Finch-Fletchley, and Hermione Granger, as their blood statuses are explicitly mentioned. The other two victims were Penelope Clearwater and Nearly Headless Nick, but it is not clear which one of the is the fourth Mudblood he is referring to.

“Haven’t you guessed yet, Harry Potter?’ said Riddle softly. ‘Ginny Weasley opened the Chamber of Secrets. She strangled the school roosters and daubed threatening messages on the walls. She set the serpent of Slytherin on four Mudbloods, and the Squib’s cat.”
_{^{- Harry Potter and the Chamber of Secrets, Chapter 17 (The Heir of Slytherin)}}

However, the main target of that basilisk attack was Hermione, because Tom Riddle knew he would be able to trap Harry much more easily if one of his best friends was attacked.

“So the foolish little brat waited until your dormitory was deserted and stole it back. But I knew what I must do. It was clear to me that you were on the trail of Slytherin’s heir. From everything Ginny had told me about you, I knew you would go to any lengths to solve the mystery – particularly if one of your best friends was attacked.”
_{^{- Harry Potter and the Chamber of Secrets, Chapter 17 (The Heir of Slytherin)}}

Nearly Headless Nick was also attacked alongside a confirmed Mudblood, Justin Finch-Fletchley, so it is also unclear whether he was intentionally targeted or not. From the information given in the books, it is not clear what either of the remaining two basilisk victims’ blood statuses are, so it is not possible to know whether Penelope was a half-blood or if Hermione did not match the name she claimed was hers with the correct blood status of its true owner.

But anything is better than the truth.

However, even a bad lie is still less likely to get Hermione in trouble than the truth of her identity. Hermione had a bounty out on her capture. Her name and picture were in the Daily Prophet, where she is described as the Mudblood traveling with Harry Potter.

““’Ermione Granger,”’ Scabior was saying, ‘“the Mudblood who is known to be travelling with ’Arry Potter.””
_{^{- Harry Potter and the Deathly Hallows, Chapter 23 (Malfoy Manor)}}

Even claiming to be someone who was also actually a Mudblood would be better than the truth that she was the Hermione Granger wanted for traveling with Harry Potter, whose capture was highly desired.

Penelope clearwater is a half-blood, not a muggle-born. So therefore, it would be better if Hermione said she was her. Also Hermione was wanted and if she said "I'm Hermione Granger", they would have brought her to Malfoy Manor straight away.