The inmate registration or ID number is what facilities use to route deposits, mail, and phone calls to the correct person. There are a few straightforward ways to find it.
If you know the state where your cousin is incarcerated, go directly to that state's Department of Corrections website and use their public inmate search. Most state DOC sites let you search by full legal name and date of birth and will return the inmate ID number along with current facility and other basic information.
If she is in a county jail rather than a state prison, search the county's online jail roster the same way. Many counties post this information publicly.
You can also use InmateAid's inmate locator, which pulls from multiple state and federal databases. If you know the state and have her full name and date of birth, the search should return what you need.
If online searches are not returning results because she was recently booked or the database has not updated yet, call the facility directly and ask the records office or intake desk. Give them her full legal name and date of birth and they can confirm her ID number over the phone.
Once you have the number, the money transfer process should be straightforward. Have her full name, the ID number, and the facility name ready when you complete the transaction.