Tzav
There are two types of dedications a person may donate to the Beis Hamikdash. Either a person may give objects which have a monetary value and these would be used to pay for the overhead and constant upkeep of the Temple. Alternatively one could offer another type of donation, these were items which could be offered on the Mizbach itself, such as animal Korbanos and grain Menachos. The more elevated of these two levels is bringing an offering on the Altar – Kodshei Mizbach. Sacrifices are intrinsically holy, as opposed to other gifts which have a monetizing value.
Question
The name reserved for a flour offering – Mincha – is most intriguing. The word Mincha translates as gift. What is unique about grain offerings that they should be identified as gifts? On the contrary shouldn’t this benevolent title be bestowed on the more costly animals rather than inexpensive grains.
Answer
While only the rich can bring animals a larger spectrum of people will be able to contribute a Mincha. Being from grain, a staple commodity, it is within the financial range of both the wealthy and the poor. Lets look at these two extremes, and their attitude towards a Mincha.
The Rich: Generally speaking it is difficult to part with a substantial amount of money, it is far less demanding to give a lesser amount than a larger one. It would be safe to assume, that the well-to-do would have an easier time giving a modest flour offering than a pricey bull offering. Thus, there would be more passion invested in the lesser gift. Perhaps there may be less quantitative value but there will be more qualitative value.
The Poor. The destitute man who is inspired to bring a offering to the Temple, would have to carefully plan his donation. Just as he has to budget his meals, he would have to organize his finances in order to bring his contribution. By saving penny after penny to collect sufficient funds he would see his aspiration materialize and be finally ready to bring his Korban. There was definitely a larger dose of emotive fervor mixed into this cheap meal offering.
It is precisely because this offering is diminutive and not too costly that it is called “Gift”. When offering a sacrifice or giving Tzedaka the most important factor is not how much it hurts the pocket of the benefactor. This is peripheral, because Hashem owns all the money of the world “Mine is the silver Mine is the gold” (Chaggai 2:8). What a person can offer is only his free will – his heart and resolve to give.
Now, regarding a bird offering, it says “a pleasing fragrance” and regarding animals, it similarly says, “a pleasing fragrance”. From here we see that both in the case of a large animal or a small bird, the fragrance is equally pleasing to G-d. This teaches us, Whether one offers much or little, it is provides identical pleasure to G-d, contingent on that he directs his heart to Heaven. (Rashi Vayikra 1:17 quoting Toras Kohanim 1:91)
The grain offering is the “Gift”. It is the Korban which is saturated with altruism. The rich will give graciously, imbuing a greater measure of feeling, and the poor put their very soul into this offering. This is the ultimate Mincha.
Story
Reb Boruch Ber Leibowitz (1864 – 1939) was the primary student of Rabbi Chaim Brisker, and was famed for his in depth Talmudic lectures. In 1904 he was appointed head of the Yeshiva in Slobodka. During the first World War he had to leave Slabodka and the Yeshiva relocated to several Eastern European cities. In 1926 he re-established the Yeshiva in Kamenitz, where it continued to attract hundreds of students for the next 13 years.
During his tenure is Kaminetz a woman unfortunately had a sick child. She came to the Rosh Hayeshiva so that he should supplicate for her family. Not being of much means, the woman gave a pitiful donation of just a few pennies so that merit of Tzeddoko should accompany the Rabbi’s merciful prayers to aid her ailing child.
Reb Boruch Ber treated the donation with the utmost respect, and hastily called for his son-in-law Reb Reuvain Grozovsky (1886 – 1958). “Please record in the ledger” he told Reb Reuvain, “that the lady has made a donation to the Yeshiva”.
Table Torah 
R’ Apter: You write, “It would be safe to assume, that the well-to-do would have an easier time giving a modest flour offering than a pricey bull offering. Thus, there would be more passion invested in the lesser gift. Perhaps there may be less quantitative value but there will be more qualitative value.” This appears counter-intuitive to me. It would seem that the easier it is for the rich man to give, the less passion he would invest due to the ease and therefore relative insignificance to him of the offering. Wouldn’t he invest more passion with the more significant gift (an animal), which cost him more? Why, because “less quantitative value,” would there be “more qualitative value”?
Wishing you a good Shabbos.
Your point is valid. I think there are three positions in the rich man’s mind. One as you point out that the donation is insignificant and does not register any passion. At the other extreme there is a gift so costly that he inwardly pains him to part with his wealth. There is a middle ground where he invests thought without feeling hurt.