Bypass the t-stat with piece of wire to make sure t-stat works before digging into further.
Check for loose connection to thermostat or system and clean contact points on t-stat.
My old mechanical t-stat likes clean contact points.
If t-stat is problem, before buying a new t-sat clean up the contact points one of two ways.
1. Remember exactly "anticipator" is before doing any cleaning. So you can put back in exact place, just in case you do move it.
2. "burnish" with stiff piece of paper (index card, match book cover etc)
3. If t-stat contacts has little nipple thing that you can't burnish without destroying nipple use a elec contact cleaner
http://www.amazon.com/gp/product/B000BXOGNI/ref=pd_lpo_sbs_dp_ss_2?pf_rd_p=1944687682&pf_rd_s=lpo-top-stripe-1&pf_rd_t=201&pf_rd_i=B003NTQCA2&pf_rd_m=ATVPDKIKX0DER&pf_rd_r=1Z7S124V0C9CDJXZB366 or if nobody is looking I use "brake clean" from any auto part store, just a tab of spray on a Q-tip to clean.