Acc. to the 教育部國語辭典, a 鼎 ding was originally a cooking pot with three feet (there were also 方鼎 with four feet), and in the Minnan dialect the word ding is still used for cooking pot. Later, however, ding came to describe the massive cauldrons used in ritual ceremonies during the Zhou dynasty.
The word ding has many connotations derived from this sense, such as 傳國的寶器 a precious vessel passed on as a symbol of royal authority, or 比喻三公、宰相、重臣等高位 a metaphor for the highest authorities of the state.
鼎力 is a very formal literary phrase that means 'with strong support', and is used in idioms like 鼎力相助, 鼎力玉成. It doesn't have to be 'state', support but it does imply that the support comes from powerful or influential persons, and is thus a flowery sort of compliment in the idioms above.