In the account of when Eliezer found Rivka to be the wife of Yitzchak, I find a passuk that could seem to be unnecessary.

At the beginning of Eliezer's speech to Lavan he introduces himself as Avraham's servant, and then says,

"G-d granted my master a very great blessing, and he prospered. He gave to him sheep, cattle, silver, gold, slaves, slavegirls, camels and donkeys."

Only then does he mention the purpose of his visit, i.e. that G-d had also miraculously granted Avraham a son from Sarah after she had grown old, and he had come to find a wife for Yitzchak.

I find the reference to Avraham's material possessions puzzling. Surely it would have been enough to have mentioned the miraculous birth of Yitzchak to persuade Rivka's household that she was 'the chosen one'. Furthermore, both Lavan and Betuel both seemingly understand the spirituality surrounding this event when they reply, "The matter is from HaShem - we cannot say anything to you bad or good." In other words, whatever their personal thoughts, hey understood that Rivka had been chosen by HaShem to be Yitchak's wife.

Again, since the spirituality surrounding the event is obvious to all, why the reference to Avraham's material possessions?

Perhaps Eliezer realised Lavan's true nature, that he was only interested in the money. In verse 29 we are told that "Lavan ran outside" and Rashi tells us this is because Lavan said to himself "This one is rich" and he wanted to see the riches for himself.

I had thought about that, and I agree that Lavan's intentions were monetary.

In this case, why did Eliezer stress the material items first and the spiritual second. He could have said that Avraham had been blessed with Yitzchak, and then followed it by saying that HaShem had also blessed Avraham with material wealth, to reinforce the shidduch as it were.

The way I read it, it could be interpreted that Eliezer was trying to emphasise the material blessings over the spiritual ones. This may be for the reason that you mention, but since Lavan's response showed that he was aware of HaShem's providence, was there any need?

When Lavan or Besuel say that matter is from Hashem and who are they to say anything, I find this to likely be insincere on their part -- to read into it how a believing Jew would mean it is probably being overly optimistic.

Perhaps they recognized that Eliezer was a believing righteous person and therefore wished to speak with him in his language.... with Eliezer comfortable they are more likely to be able to poison him and steal his posessions, or otherwise be in control of the situation.

This doesn't address your main point of why mention both types of special kindnesses Hashem gave to Avraham to convince them that Rivka was Yitzchak's chosen wife.

I am not even sure the words convey such confidence in that matter... Eliezer says if they do not agree he'll happily move on to elsewhere to find a wife for Yitzchak.

Attention grabber maybe?

As previously stated, Lavan's truest interest is in material wealth- so this would grab his attention to be open to listening to the shiddich in the first place. Lavan wasn't a person concerned with the spirituality aspect as much as the materialistic. (Although, he DID have his own concerns with "spirituality"- as he was an idol worshipper.)

Every good advertisment needs to speak to the buyer's desires.