If you’ve only got an account name you're going to have to do some work! It depends on what’s happened to the account. If you’ve let the account lapse for a long time the bank should have written to you to ask what you want to do with it.
If you’ve moved house and forgot to tell the bank your new address and their letters get no reply - they just stop sending reminder letters and your account will be marked as ‘lost’
This doesn’t mean it’s really lost - just lost till you claim it!
There is a good site in the UK called lost account which helps people trace savings and current bank accounts. It’s free and they can help -even if the bank has closed or merged with another and you don’t remember what it’s called
Most of the banks are in the scheme and even some foreign banks are included if they operate in the UK.
You have to complete the claim online and fill in as much detail as you remember.
They have a similar site in the USA called missing money. They don’t do the search for you but they make it easy to do it yourself.
You should never pay for this in either country as these are government schemes.
It's a bit time consuming, but both sites have good FAQ pages and it's worth doing, especially if you can get your hands on some money!