I am extremely sure of the meaning as it is a prayer I often recite. The expression: Changer of the hearts is an appositional noun of Lord. So it is as if you are calling him by two names: the 1st: Oh Lord! the 2nd: Oh Changer of the hearts. The reason for this second name is that God is the One who determines how you feel, the stae of your heart. Finally the expression ثَبِّتْ is an imperative of ثبَّت، يُثَبِّتُ to fasten. However, in prayer we do not call it an imperative. It is called a petition. The last thing I want to say is that the prayer in itself is very poetic as قَلِّبْ and قُلُوب are both from the root ق ل ب .

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Note added at 42 mins (2006-05-25 15:38:35 GMT)
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And as they are from the same root, they sound alike.

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Note added at 1 hr (2006-05-25 16:28:12 GMT)
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Changer of hearts sounds even better than Changer of the hearts.

another option :)

Slaves of Allaah! This supplication which reflects humility and humbleness to Allaah, was the supplication of the best of creation, our Prophet sallallaahu ‘alaihi wa sallam. Uttering this supplication is an indication of perfect imitation of the Prophet sallallaahu ‘alaihi wa sallam, as ‘Um Salamah, may Allaah be pleased with her, said, ‘The Prophet sallallaahu ‘alaihi wa sallam used to repeatedly say this supplication, “O Controller of the hearts! Make my heart steadfast upon Your religion” so I asked him, ‘O Prophet of Allaah! Do hearts fluctuate?’ He said, “Yes, all mankind’s hearts are between two of the fingers of Allaah. If Allaah wills He maintains a man’s steadfastness, and if He wills He deviates him” (Ahmad & Tirmithi).

http://www.alminbar.com/khutbaheng/2600.htm

Or if individuality is a must:
"O Lord, who changes man's heart, give me a steadfast heart."

"changer of the hearts" can be interpretted as "he who changes haearts (i.e. who changes his mind) frequently", which is a disaster!

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From the 13th century prayer of Thomas Aquinas.

"Give us, O Lord, steadfast hearts,
which no unworthy thought can drag downward;
unconquered hearts,
which no tribulation can wear out;
upright hearts,
which no unworthy purpose may tempt aside.
Bestow upon us also, O Lord our God,
understanding to know you,
diligence to seek you
wisdom to find you,
and a faithfulness that may finally embrace you"

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Note added at 6 hrs (2006-05-25 21:20:34 GMT)
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"O Lord, who changes man's heart, keep my heart on yuor faith."

Unless you seek abstraction, I recommend Zeinab's answer.

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Note added at 6 hrs (2006-05-25 21:21:21 GMT)
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while considering my comments of course.

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Note added at 6 hrs (2006-05-25 21:23:33 GMT)
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*typo:
"(...)on your Faith."

Also:
O' Allah, the Turner of the hearts, keep our hearts turned to Your (or Thine) obedience