Well let's see here,
If you're looking for a backronym in Tanach it's pretty slim pickings inspiration-wise.
Ezekiel 7:20:
וצבי עדיו לגאון שמהו וצלמי תועבתם שקוציהם עשו בו על כן נתתיו להם לנדה
and statues of their awful, disgusting things they made ...
Isaiah 38:12-13:
מיום עד-לילה תשלימני. שויתי עד-בקר כארי
I was a goner; I braced myself till morning.
I'm not finding any sofei teivos.
Moseying on over to the Babylonian Talmud, we do a bit better. In order of increasing inspiration (well according to me anyway):
Temurah 3a:
תעשה שבתורה עשה בו
A "don't do" commandment, if [actively] done
Okay! I'll try not to sin in 5772! What about working on my relationships with fellow man?
Kiddushin 63a:
תניא שב עמי בצוותא
It was taught, "sit together with me."
Eruvin 48a:
תימא שנתנו עירובן באמצעית
If you say they put their eruv in the middle
Alright; how about some Torah study for the new year?
Sanhedrin 44b:
תמידין שנאמר עתה באתי
[Torah study is greater than] daily sacrifices, as it says [in Joshua], "now I came."
And finally, now shouldn't I pay more attention to prayer this year?
Megillah 17b:
תיקנו שמונה עשרה ברכות
[The sages] enacted 18 blessings
If we go for Gematrias, Jeremiah 1:17 is a complete verse with Gematria 5772, and is pretty inspiring:
ואתה תאזר מתניך וקמת ודברת אליהם את כל אשר אנכי אצוך אל תחת מפניהם פן אחתך לפניהם
Now you shall gird your loins, get up and tell them what I command you; don't break with fear of them or I will break you with fear in front of them.
There's actually one more complete 5772 verse, but a bit more depressing:
ויקשר קשר הושע בן אלה על פקח בן רמליהו ויכהו וימיתהו וימלך תחתיו בשנת עשרים ליותם בן עזיה
About one Jewish king who overthrew and killed another one. Woo-hoo. (This was in the "circling the drain" period for Israel's North Kingdom.)
There are plenty of Biblical phrases with gematria 5772, but often they're sentence fragments that leave you hanging; this one works pretty well on its own:
כאשר שש על אבתיך כי תשמע בקול יהוה אלהיך לשמר מצותיו וחקתיו הכתובה בספר התורה