Nothing you (or the author) can do to the widget can determine whether your router can get data from Accuweather, and nothing you can do to the router can affect the widget.
What I suspect (and this is a wild guess*) is that the Accuweather widget uses Accuweather's canonical address (the human readable address, like accuweather.com). The router gets that request from the phone and asks the DNS server it's using (probably your internet provider's) for the numeric address for that canonical name (220.127.116.11 is accuweather's). Canonical names are just for us stupid humans, who can't remember a few thousand strings of numerical addresses.
If the router loses the DNS server's address (or, more likely, since so many kids who have too much time on their hands know more than is good for them, someone poisoned [basically changed] the address in that server), any app that uses a numerical address - since it already has the numerical address, it won't ask for it - gets to the server it needs and everything works. But an app that uses a canonical address gets the wrong numeric address, doesn't get the data it needs and, rather than telling you that the temperature is "a left turn at Broadway", it doesn't change anything on the screen.
*The reason I said "wild guess" is that if, when you're having this problem (and you're absolutely certain that you're connected to YOUR router, not to a neighbor's wifi signal) you can open your browser and get to a website by name (like androidforums.com - I usually use aol.com because it's easier to type and because I can't remember the last time they were down long enough to get dropped from the DNS system - then that may or may not be the problem.
But a) rebooting the router may get a different DNS server, if your ISP uses a few, your router's DNS cache may have gotten poisoned or courrupted, and rebooting fixes that and b) your phone's data connection uses your phone company's DNS server, so a wifi DNS problem won't show up on your data connection.
Next time it happens, check your wifi connection - make sure it's still yours. (If not, fix that, tap the widget and it should open the app with the current conditions, and also update the widget.)
If it is, connect (on a laptop or desktop) to accuweather.com and see if they're down. (It could snow in July in Miami too.) If not, go to AccuWeather.com - Contact Us
, tell them what's happening (there's a contact form) and ask whether it could be a DNS problem - IOW, are they going by a canonical address, so you should contact your ISP when it happens, or are they using a numerical address.
If they're using a canonical address, ask if you could get the exact one, so your ISP can check for poisoning. Their tech people will understand what that means. They'll probably give you an address - accuweather.com/<something here> or something like that.
Then, when it happens, call your internet provider's tech support number (hope it's better than mine - I reamed them about a dozen new ones this afternoon [I TOLD YOU the location of the store, I asked you for its PHONE NUMBER (which isn't listed on your site), so don't give me a list of the addresses of all your local stores - without phone numbers]), explain the situation and tell them that the particular address - and you have it - may have gotten clobbered in their DNS servers, and could they check it. If they're any good, they'll take the address from you, verify that it's poisoned and fix it up. (If they're like mine, they'll pretend to understand, tell you that they'll get it fixed, then go for coffee. Unfortunately, there aren't too many choices around here, and the "other" cable company has a yearly budget that I blow at Walmart for groceries.)
(Just had another thought. If the router is really getting old, you might want to think about replacing it. The DNS cache could be getting corrupted just due to things in the router breaking down. My main router is new (the old one finally gave it up after about 10 years), but the spare one is 12 years old and still works [even after being left out in the woods one winter - looks like a mess, but works]. When the main one went, I used the spare for a couple of days, until I had time to go out and buy a new one. Wifi was a bit slow, but internet cold turkey? I think the internet is more addicting than cocaine. And the house phone is VoIP, so I need internet.)