I know that I can use a tool like NMAP or arp-scan on Linux to identify the IP and MAC addresses of all devices on my local network. I also know that arp-scan will do a MAC address lookup to get the device manufacturer. But is there any set of options on these commands (or any other command) that can tell me the actual device name of the device at a give IP? For example, if "Joe's iPad" is on 192.168.1.113 I want a command to get that name.

Nmap

Some hosts could simply be configured to not share that information. It should work just like this:

user@host:~$ nmap 192.168.1.113



Starting Nmap 7.00 ( https://nmap.org ) at 2015-12-11 08:45 AWST

Nmap scan report for Joes iPad (192.168.1.113)

Host is up (0.0038s latency).



Not shown: 999 closed ports

PORT      STATE SERVICE

62078/tcp open  iphone-sync



Nmap done: 1 IP address (1 host up) scanned in 41.88 seconds

You can force Nmap to attempt reverse DNS resolution for all targets, by using the following option:

-R (DNS resolution for all targets).



    Tells Nmap to always do reverse DNS resolution on the target

    IP addresses. Normally reverse DNS is only performed against

    responsive (online) hosts.

This might help in some cases. The output will look more or less identical:

user@host:~$ nmap -R 192.168.1.113



Starting Nmap 7.00 ( https://nmap.org ) at 2015-12-11 08:46 AWST

Nmap scan report for joes-ipad.local (192.168.1.113)

Host is up (0.0047s latency).

rDNS record for 192.168.1.113: joes-ipad.local

Not shown: 999 closed ports

PORT      STATE SERVICE

62078/tcp open  iphone-sync



Nmap done: 1 IP address (1 host up) scanned in 42.61 seconds

Either way; you can always parse the output through external tools, such as grep. This can be particularly useful if you're scanning several addresses, or even whole network ranges at a time:

user@host:~$ nmap 192.168.1.0/24 | grep '(192.168.1.113)'



Nmap scan report for Joes iPad (192.168.1.113)

All 1000 scanned ports on Joes iPad (192.168.1.113) are closed

user@host:~$ nmap -R 192.168.1.0/24 | grep '(192.168.1.113)' 



Nmap scan report for Joes iPad (192.168.1.113)

All 1000 scanned ports on Joes iPad (192.168.1.133) are closed

Net-Tools

What you (probably) actually want to do, is this:
^{*Output will vary, based on OS & software version.}

user@gnu:~$ arp 192.168.1.113



Address                  HWtype  HWaddress           Flags Mask            Iface

Joes iPad                ether   a1:b2:c3:d4:e5:f6   C                     wlan0

user@bsd:~$ arp 192.168.1.113



Joes iPad  (192.168.1.113)  at  a1:b2:c3:d4:e5:f6  on   en0  ifscope  [ethernet]

Take a look at nmblookup and/or smbutil lookup commands.