You wouldn't ask someone to sing in that particular style if you just wanted to be reminded of the melody. You would go with something more neutral. "Sing" or "hum" sound good in the blank, but "croon" is too markedly specific.
As an aide-memoire it might be worth pointing out that there is a very well-known phrase:
You hum it, I'll play it.
This is typically said by a pianist in a bar or restaurant when a customer requests a tune and the pianist doesn't recognize the title. If the customer hums the tune then the pianist will be able to pick it up.
Note (Not too off-topic I hope)
There is also a well-known joke based on the phrase. If you search online for You hum it, I'll play it followed by the word 'joke' you'll find examples.