In Parashas Chayei Sarah, the Torah learns us that the servant of Avraham took ten camels when he departed on his mission. Rashi explains what this means:
He wrote a deed of gift of all his possessions in favour of Isaac so that they would be eager to send him their daughter
Rashi got this idea, that Avraham wrote a deed of gift of all his possessions, from a Midrash Rabbah. See also the Be'er Mayim Chayim.
The Lubavitcher Rebbe explains this concept in great detail (Sefer HaSichos 5752, p 103) (see: Gutnick Gumash).
"Avraham had a very large amount of wealth, as recounted in earlier
passages etc., surely even if Avraham wrote a small portion of his
wealth, that would be more than enough to impress the parents etc.?"
The Rebbe begins to cite a Pirkei D'Rabbi Eliezer, which says that Avraham Avinu gave Eliezer a will stating that he had bequeathed everything to Yitzchak. He then goes on to explain why Rashi reject this view of the Midrash, saying:
Rashi was troubled by an implicit contradiction in verse 10: "In his hand was all his master's belongings". Surely, the very definition of a master is a person who is rich in belongings? If Avraham had indeed given all his belongings away to Eliezer, then in what respect was he still a master over Eliezer? [...] To solve this, Rashi answered that Avraham gave his possesions, not to Eliezer, but to Yitzchak.
The Rebbe then goes on to build further on the idea why Rashi rejects the Midrashic idea that a will was given:
Since this would not have made an immediate impression on Lavan and Besu'el, since it was likely that many years would pass before Yitzchak would inherit the money. Similary, Avraham gave Eliezer all his possesions, in order to guarantee as speedy a response as possible from Lavan's family. As for Avraham's own financial situation, we could presume that either: a) Yitzchak supported him, thus fulfilling the mitzvah of honoring one's parents, or, b) He sought means of making money.
Nachmanides, the Ramban, explains in his commentary on Bereishis 24:10 that "all the goods of his master in his hand" refers to:
All the best and superior kinds [of food] - fruits and delicacies- from all that was found in Damascus
So, the Ramban seems to say that it does not refer to money, but to the choicest of foods of that geographical location, Damascus.
The Or HaChaim, in his commentary explains a familiair thing as the Ramban, but does not connect it to food:
וכל טוב אדוניו, "and all the good things of his master," - Eliezer took the choicest of all the precious possessions of Abraham.
Another fascinating insight, this time a kabbalistic one, is given by the Shelah (Torah Shebikhtav, Chayei Sara, Torah Ohr 54):
When the Torah (Genesis 24,10) describes Eliezer as having כל טוב אדוניו בידו, "all the good of his master in his hand," this refers to the angel Mattatron who accompanied him.
Finally, you ask:
So why did Avraham give "everything he had" for this mission?
The Or HaChaim explains:
This was in order to persuade the girl in the event that she would initially refuse to move away from her parental home. Seeing that her husband-to-be possessed such wealth might change her mind.
There are a few midrashic mefarshim (Etz Yosef and Perush Maharzu) that discuss why all posessions needed to be given. It was so that they would send their daughter.