The main things to take into consideration:
- Some banks offer actual savings accounts in the child's name, while some require them to be custodial accounts (where the parent must sign off on withdrawals). Those may be called UTMA, UGMA, Custodial, or ITF accounts. Each have benefits and drawbacks depending on how you want your child to be able to interact with their bank.
- Online versus in person. Some prefer online banks (which tend to give better interest rates, and allow more flexibility with choosing features right for you) while some prefer a physical bank to give your child the feeling of banking. This may depend in part on your physical location - if you're far from a "good" bank option, go with online.
- Watch out for fees, particularly monthly maintenance fees. Find a bank that doesn't charge them for children. Interest may not be a big deal at this point, but fees are.
If you're looking for an online bank, most of the big online banks have child savings account options, though they vary significantly in what features they offer. Ally has a UTMA account, for example, while Capital One has both a kids' savings account and a Teen Money account which comes with a debit card. Find one that has the features you need.
Most of the online banks have very good online app access (though, again, double-check!), and many of the larger in-person banks also have good apps that let you see your transactions. Chase, Capital One, Discover all have fairly good apps for that purpose.