What is the minimum interest income that must be claimed?

You have to report all your income, no matter how little it is, whether or not you get a 1099-series reporting form.

I'm not sure what you're referencing, but:

• If a bank, financial institution, or other entity pays you at least $10 of interest during the year, it is required to prepare a Form 1099-INT.
• There is no reporting minimum for an individual.  All 1099-INT should be included on your tax return.
• If your child's only income is interest and dividend income (including capital gain distributions) and totals less than $11,500, you may be able to elect to include that income on your return rather than file a return for your child. Tax on a Child's Investment.
• If this was not your question, please give us a few more details.