There isn’t a particular promise that is referred to in Psalm 119:76. But a general promise that God will act according to His lovingkindness.
Even though the New International Version (NIV) translates the word to promise, another translation is from the New American Standard Bible (NASB1995), which translates the same verse to the following:
“O may Your lovingkindness comfort me,
According to Your word to Your servant.” (Psalm 119:76)
The NASB is a more literal word-for-word translation from Hebrew than the NIV (which is a phrase-by-phrase translation for better readability). So the actual word used here is “Your word” instead of “Your promise”.
However, this is a minor point.
Verse 76 in Psalm 119 explains that God promised beforehand to comfort His servants with lovingkindness. We can all expect God to comfort us with His lovingkindness if we serve Him.
There are many verses in the Bible of how God will comfort us.
“For the Lord is good;
His lovingkindness is everlasting
And His faithfulness to all generations.” (Psalm 100:5)
“You will make known to me the path of life;
In Your presence is fullness of joy;
In Your right hand there are pleasures forever.” (Psalm 16:11)
“Many, O Lord my God, are the wonders which You have done,
And Your thoughts toward us;
There is none to compare with You.” (Psalm 40:5)
“For the Lord God is a sun and shield;
The Lord gives grace and glory;
No good thing does He withhold from those who walk uprightly.” (Psalm 84:11)
God promised us a Savior to save us from sins.
"For a child will be born to us, a son will be given to us;
And the government will rest on His shoulders;
And His name will be called Wonderful Counselor,
Mighty God, Eternal Father, Prince of Peace." (Isaiah 9:6)
Through this Savior, we will have eternal life to those who believe in Him:
“For God so loved the world, that He gave His only begotten Son, that whoever believes in Him shall not perish, but have eternal life. For God did not send the Son into the world to judge the world, but that the world might be saved through Him. He who believes in Him is not judged; he who does not believe has been judged already, because he has not believed in the name of the only begotten Son of God.” (John 3:16-18)
And those who serve the Son of God (Jesus), the Father will honor:
“If anyone serves Me, he must follow Me; and where I am, there My servant will be also; if anyone serves Me, the Father will honor him.” (John 12:26)
“Things which eye has not seen and ear has not heard,
And which have not entered the heart of man,
All that God has prepared for those who love Him.” (1 Corinthians 2:9)