Okay. It's going to be a little harder to diagnose it since the problem isn't immediately visible, but you've got all the Linux toolset there, so that's helpful.
Is the DNS server you're trying to use from the LAN machines running on the OpenWrt machine, or off somewhere on the Internet?
EDIT: Or on the LAN, I guess.
EDIT2: Oh, you answered that elsewhere.
I am using my routers DNS, and it’s reachable from my laptop.
Have you tried doing a DNS lookup from the router (pinging a host by name, say) when you were having the problems?
If so and it didn't work, that'd suggest that the problem is the upstream DNS server. If that's the problem, as IsoKiero suggests, you might set the OpenWrt box to use a different DNS server.
If so, and it worked, that'd suggest that the issue is the OpenWrt host's DNS server serving names. It sounds like OpenWrt uses dnsmasq by default.
If not...that'd probably be what I'd try next time the issue comes up.