Unable To Resolve URL?

My mom has an IBM PC which recently had to have the HD replaced. When we got it back we had a brand new HD with a new install of XP Professional with SP 2 (old HD had XP Home edition only).
The computer connects to the internet via a dial up link only. The computer works fine except that I notice after we have been on line for a time e.g. > 1hr or so, all of a sudden with no warning we seem to loose ability to resolve URL names when trying to connect to websites. In Firefox we get the message in the bottom of the Firefox display ''looking up www.abcde.com'' and then after a while the message that the web site can't be found. IE has a similar issue and appears unable to resolve the URLs to IP addresses and trying to ping a URL (e.g. ping www.yahoo.com) produces an error message indicating the address can't be found.
I checked the IP config info and when this problem occurrs I find I am still able to ping the DNS server's IP address without a problem from the computer (so the basic connectivity appears to be still there), yet I'll still get the message in browser in the end indicating name resolution failed. When I check the stats on the connection properties while I am trying to connect to a web site the bytes sent/recieved counters do increment indicating something is going back and forth between the computer and the ISP and no excessive amounts of errors are reported.
Sometimes the problem occurs after an hour or so of surfing, other times we'll be connected and surfing without a problem, but we logoff for a while leaving the computer running and then come back, logon and find this name resolution problem has cropped up again and it's impossible to resolve URLs again until a reboot.
The only fix is a reboot. Logging off and logging on and/or disconneting/reconnecting the dial up link has no effect unless a reboot is performed first. Once we reboot, we can connect and surf again with no problem.
On the computer we have one account set up with admin privelidges and 2 user accounts. The problem occurs on all 3 accounts and once the problem starts occurring changing accounts doesn't make a difference (without a reboot).
I checked the system log for error messages while this problem was occuring and nothing showed up in the system log.
I would appreciate any suggestions as to where I should look next. This problem only occurred after we got the computer back with the new hard drive and new OS installed, prior to that the problem never occurred on the old hard drive with XP Home edition. I was thinking of looking for a sniffer freeware/shareware program to see if we could determine what exactly was going out over the wire and coming back when this problem starts to occur, but since we only have dial up access downloading software is a pain.