Autodialler its called and yes you can do just that on contact closures. We have these at some remote locations to signal Mains failure and intruder alarms etc being set off.
A timer could do just that but make sure its battery backed in case the mains packs up;)..
There is a very nice Menvier one that can signal you if the temp drops too low ideal for checking that Granny is OK and not dying of Hypothermia if the central heating packs up at night etc...
This is the Granny version you can even listen into what's going on there;!..
BTW why don't you get your phone service of someone like Zen Internet they are a bit cheaper and I don't think they have the odd "must make calls" malarkey either..
And theres none of that connection charge bollox either..