When I access a linux machine with sshfs, I take the following steps.
Can I use a name, not ip address to access the linux machine?
To me, avahi seems to be the best solution as it enables Mac understands the name without any configuration. With ubuntu, I could install avahi as follows :
sudo apt-get install avahi-daemon
Assuming that the machines are on the same network, you should be able to just use the machine's host name instead of its IP address.
What you want is the resolution from a network name to an ip address, which is exactly what DNS is for.
If the linux machine is on your LAN, IMHO dnsmasq would fit best.
Dnsmasq is a lightweight, easy to configure DNS forwarder and DHCP server. It is designed to provide DNS and, optionally, DHCP, to a small network. It can serve the names of local machines which are not in the global DNS. The DHCP server integrates with the DNS server and allows machines with DHCP-allocated addresses to appear in the DNS with names configured either in each host or in a central configuration file. Dnsmasq supports static and dynamic DHCP leases and BOOTP/TFTP/PXE for network booting of diskless machines.
If its on the internet, you could use some dynamic dns provider. A popular free provider is DynDNS.
prosseek-home.dyndns.org, you can give it a nickname in ssh with a Host home  HostName prosseek-home.dyndns.org  section in ~/.ssh/config.
There are several possibilities.
/etc/hosts (only for static IP address)