The following is based on discussions I had with my chevruta, DE.
The gemara on 4B in Mesekhet B'rachot:
אמר רבי אלעזר א"ר אבינא כל האומר (תהילים קמה) תהלה לדוד בכל יום שלש פעמים מובטח לו שהוא בן העולם הבא
מאי טעמא אילימא משום דאתיא באל"ף בי"ת נימא (תהילים קיט) אשרי תמימי דרך דאתיא בתמניא אפין
אלא משום דאית ביה (תהילים קמה) פותח את ידך נימא הלל הגדול דכתיב ביה (תהילים קלו) נותן לחם לכל בשר
אלא משום דאית ביה תרתי אמר רבי יוחנן מפני מה לא נאמר נו"ן באשרי מפני שיש בה מפלתן של שונאי ישראל דכתיב (עמוס ה) נפלה לא תוסיף קום בתולת ישראל במערבא מתרצי לה הכי נפלה ולא תוסיף לנפול עוד קום בתולת ישראל
אמר רב נחמן בר יצחק אפילו הכי חזר דוד וסמכן ברוח הקדש שנא' (תהילים קמה) סומך ה' לכל הנופלים:
The gemara states that one who says ashrei three times a day is guaranteed a place in olam haBa. The gemara explains that this is because ashrei has two important features: 1) the first letter of each verse of ashrei forms an acrostic of the alef-bet; 2) it contains the verse, "poteach et yadecha" which expresses the idea that God provides us with our food (=or needs). There are other chapters of Tehillim that have these features but only ashrei has both.
I would like to take up two issues:
1) In the course of the gemara's discussion it says that the 119th chapter of Tehillim is superior to ashrei - in regards to the first mentioned feature - because each letter of the alef-bet is represented eight times (i.e., eight verses for alef, eight verse for bet, and so on).
Why would it be superior for each letter of the alef-bet to be repeated eight times? Even more importantly, what is the significance of a chapter of Tehillim forming an acrostic of the alef-bet?
2) Why would one be guaranteed a place in olam haBa for saying ashrei three times a day?
In Tehillim 106 it is written: "מי ימלל גבורות ה' ישמיע כל תהלתו" - "Who can relate the mighty deeds of G' - make known all of His praise?" This is one of the most basic principles of tefilla. There is a limit to how much we can praise G' for one simple reason: we are limited in our ability to know Him. I believe the use of the alef-bet acrostic expresses this idea. It is as if we are saying: "We have a limited tool with which to praise You, G': our speech. We will fully exhaust this capacity (by using all the letters) but, as we said, it is clearly limited, being composed of only twenty-two (chaser achat) letters."
So why is it better to repeat each letter eight times? If anything, it would seem to detract from the message of limitation. I believe the Meiri (and I think the Radak says almost the same thing1 - but I don't have access to it right now) expresses the point beautifully in his introduction to Tehillim 119. He states that he does not know what the intention of the eight-fold repetition is. However, he does offer an intriguing theory: because David HaMelech's overwhelming desire (to know G's ways/learn His Torah/serve Him) he hints at his desire to remove himself from all physical pursuits and to only involve himself in the physical in so far as it is necessary to sustain his soul; to subordinate all his physical powers to the service of G', as he indicates when he writes (Tehillim 119:10), "בְּכָל-לִבִּי דְרַשְׁתִּיךָ" - "With all my heart I seek you..." The Meiri explains how the number eight hints at this idea: the body has eight parts: the five sense; the vegetative soul; the desiring/sentient soul; the rational soul2. Now we can see why the eight-fold repetition is superior. The alef-bet expresses our desire to fully exhaust our faculty of speech in the praise of G'. The eight-fold repetition of the alef-bet goes further - it expresses our desire to fully exhaust all of our faculties in the service of G'.
So why would one be guaranteed a place in olam haBa for saying ashrei three times a day. I think the significance of three times is clear: this is the number of tefillot we say each day, our most regular form of avoda. Ashrei expresses two of the most fundamental ideas in our Avodat Hashem: 1) the limitation of our ability to praise/know G' - expressed in the alef-bet acrostic; 2) our absolute dependence on Him - expressed in the verse "poteach et yadecha". It is reasonable to say that one who truly inculcates these ideas (stating them is simple) in their Avodat Hashem is guaranteed a place in olam haBa.
1 In the הוצאת מקיצי נרדמים of the Meiri on Tehillim, Yehuda Paris mentions in his notes (he references אגרות שד"ל p. 537 - I do not have a copy of this book), that are printed after the פירוש, that much of the material in the Meiri's commentary was taken from the Radak's long commentary on Tehillim.
2 I find it strange that the Meiri and other m'farshim struggled to find the meaning of the number eight. We have many examples throughout the Torah of eight: milah; the eighth day of/after the miluim; sh'mini atzeret. - R' Leibtag discusses the significance of this number in this shiur. He says, "Thus, the eighth day represents yet one more level of progression in the creation process, which first took place in seven days. [This is what the Maharal calls "m'al ha'teva" - above nature!]" I, בע"ד, would suggest that the eighth day (or an eight-fold repetition) signifies man's role within the Creation - how man acts, or even improves upon, this creation. This explanation would do away with the necessity to count the parts of our body. It seems the Meiri and other m'farshim found eight problematic because it is not a "natural" number like seven and ten (עיין שם).