How does a pre-approved credit card offer, affect the probability of actual approval?

These credit card pre-approvals are mostly meaningless. Receiving one basically means that they deemed you worth advertising to. That's an incredibly low bar, and means almost nothing when it comes to actually being approved.

Very commonly people don't get approved for the advertised offer, but instead for a lower limit and higher interest rate.

That said, your cousin's history with the secured card makes it sound likely they'd be approved for a low limit card, but this card doesn't seem like a good option. $35 fee in the 2nd year on such little spending is not justifiable and with so little credit history it's not ideal to be closing accounts after a year. Better to find a no fee reward card if looking to build credit and benefit from rewards a little.

It means they'd love to give her $5000 of credit that she's probably not going to pay off immediately. I wouldn't count on her being rejected.

Credit card companies love giving college-age/near college-age kids a credit card because they're likely to run up a balance and make minimum payments for years to come.

Several credit card companies set up on college campuses and hand out free food for signing up. All of my friends who signed up was approved. They were 17-18-year-old college kids with no job and paying for everything with student loans. The cards are horrible, super high APR, and a fee after the first year.

She's already done the right thing in getting a secured card to limit how much financial trouble she can get into. Let this be the first lesson on trashing anything with the words "Pre-Approved" on it.