Question: Can you please tell me if it is correct and natural to say *find someone by their phone number* in the context of searching someone's information in a system? For example:

> 
> I don't see any appoinments set up under your name in our system. Let me try to **find you by your phone number**.
> 
> 
> 

If it doesn't sound natual, does *let me try to find you **with** your phone number* sounds better?

If they don't sound natural, would you give me alternative phrases to convey the message.

Answer: 
It's quite correct and natural. It's another case of an understood word 'using', which would be after the 'by'.

Possibly a better way would be 'using your phone number' (in this case the 'by' is understood.

Probably the most natural way of saying this would be simply: "Let me try your phone number".

Again, there are 'understood' words in this shorter example. The full sentence, would be: ""Let me try and find you by using your phone number", but no one would say that in practice.
One of the uses of "by" is when talking about the way things are organized:

* These books are arranged alphabetically **by** the author's last name.
* Please line up **by** height from shortest to tallest.
* This table of the wealthiest people in the world is ordered **by** their net worth.

This makes it a great choice for the example situation, in which the system can be indexed "by" phone number (as well as perhaps in other ways, like "by" the person's name). You *could* choose "with", since the phone number would help you in the task of looking up (just like "I'll paint this house **with** a paintbrush"), and it wouldn't be odd, but the "lookup" sense of "by" makes it a more likely choice.