One angle that the mefarshim explain explicitly, is that these "gifts" were not gifts in the literal sense but relating to the names of G-d or pagan deities. In other words, they came from Avraham's wider understanding of the world and the various areas of knowledge he had accrued over time.
For example, the Daas Zekeinim notes:
נתן אברהם מתנות, “Avraham had already given gifts.” According to the Talmud in tractate Sanhedrin folio 91, these “gifts” were the ability to utter the names of the holy name of the Lord without thereby incurring severe harm. Even nowadays, according to our author, the Moslems who are trained in theology are able to do this without coming to any harm thereby. It was difficult for Rabbi Moses to understand how Avraham a great and devout man could possibly have taught the holy Name of the Lord to wicked people. In view of this problem, Rabbi Yaakov, son of Rabbi Nachman, interpreted that line in the Talmud differently by saying that the word טומאה used in the Talmud there refers to the names of certain demons. The names of these demons are associated with the evil spirits making their homes in certain ruins, or certain cemeteries as described by the Talmud in tractate Chagigah folio 3. This is alluded to by the fact that the spelling of the word “matanot” is spelled as if it were a singular, the letter ו having been omitted. As spelled in the Torah here, the numerical value of the letters in the words נתן אברהם מתנת equals the numerical value of למדם להשביע השדים. [I have not quite figured this out. At any rate, the idea is that Avraham taught these sons of his how, when they used the names of deities when swearing an oath, they could do it in such a way as not to be guilty of violating the third of the Ten Commandments.
If we go with the second p'shat it would answer your question as to why Avraham didn't want to derive benefit from such gifts as they are based on impure / evil foundations.
Refer also to the Chizkuni for a similar approach.
The HaKesav Vehakabbalah explains why Avraham acted as he did:
הודיע להם שמות אלה כדי ליתן אל לבם לדעת להשתמר מהם ולא יבואו להחליף שם טומאה בטהרה ולעבדם בלתי ידיעה
He informed them of these names in order to let their hearts know to guard against them and they would not come to replace impurity with purity and serve them without knowledge.
If we build on this approach, we can answer why Rashi gives no source. They were not gifts in the literal sense (although there are some commentaries that suggest they were indeed of monetary value - e.g. Radak) and thus Avraham only had knowledge to pass on. The Sifsei Chachomim makes the logical deduction:
שם טומאה מסר להם. והא כבר נתן ליצחק כל ממונו:
He transmitted to them impure names. [Rashi knows this] because Avraham had already given all his possessions to Yitzchok.
As such, logically, it did not have to state where they came from, as they were of a tangible/material nature as it was just forwarding knowledge.