Reliance; a request or dependence on something. Often used as the suffix 〜だのみ to mean 'depending on / relying on.'
準備は君だのみだ。
I'm counting on you for the preparations.
あとは天候だのみで、試合が行えるか決まる。
After that, it's down to the weather; whether the match can be held will be decided by it.
彼は友人に頼みをした。
He asked his friend for a favor.