I found a manual for the Linksys WRT610 V1 router online. I believe you will be able to get your Slingbox working reliably on your local network if you program your router to assign a static IP address to the Slingbox. To do this, you will need the MAC address for your Slingbox which is printed on a label on the bottom of the slingbox, and also on the shipping box.
(Most routers are set up by default to dynamically assign IP addresses to every device attached to them. But these assignments can change over time, especially if you suffer power failures or bring different computers into your network after being away for a week or more. The computers themselves have no problems with being assigned different IP addresses by the router from week to week, but the Slingbox is finicky about that, since the combination of the IP address for your modem plus the local IP address for the Slingbox on your network is how the Slingbox directory knows where to find your Slingbox in Internet Land when you go to www.slingbox.com and login to your Slingbox account.)
After you have copied down the MAC address for your Slingbox, then look at Chapter 3 (Network Setup) in your Manual and follow the instructions for assigning a static IP address to a device (your Slingbox). The copy of the manual that I am reading online says your router uses the IP address http://192.168.1.1 for its base address and assigns IP addresses to equipment attached to it starting with address 192.168.1.100, and (by default), goes up to 192.168.1.149, for a total of 50 addresses that it will choose from to dynamically assign IP addresses. Your router's manual says that static IP addresses should be selected from this range of 50, while other routers I've worked with (including my own) recommending assigning a static IP address that is specifically OUTSIDE the range of dynamic address pool that the router is using. So, I would suggest following your router manual's recommendation and assign the IP address 192.168.1.149 to your Slingbox (you can use "MySlingbox" as the Client Name to enter, following the manual's instructions).
Then, go to page 25 in your manual -- the section called "Port Range Forwarding" and add an Application entry (you can call it "SlingboxPort") to the table, with starting address 192.168.1.149 and ending address 192.168.1.149. Choose protocol setting "Both". Enter 5001 as the port number that you are assigning. Click the "Enabled" box and click the "Save settings" box.
Then power cycle (i.e., simply unplug the power cord for about 30 seconds) for both the router and the Slingbox.
Plug the power back in for both router and Slingbox. When both devices boot back up, then you should be able to log in to your Slingbox with both the standalone Slingplayer software that you install to your computer and the web-brower add-on for viewing the Slingbox video within a webbrowser window.
With a static IP address now assigned to your Slingbox, your computer should always be able to find the Slingbox regardless of where you happen to be at that time in Internet Land.