The Team Benefactor is the member of the team who allows the adventure to continue by supplying some sort of resource they (and only they) have access to rather than their own talents (unless said talents are something very narrow). The resource in question might be;
Money (Either covering the group's expenses as they travel or providing a large sum of money to fund the trip).
Knowledge of what they're specifically dealing with (characters who provide this are often The Smart Guy as well). Military fiction (especially Military Science-Fiction) often has a civilian expert (or witness to whatever the protagonists are dealing with, local guide, etc) in this role.
Have something special about them (e.g. being of a given lineage needed to enter their ancestor's tomb or meeting the Phlebotinum Handling Requirements) that the group need to continue.
Just own a means of transport which the group need to make use of.
Have some sort of connection to someone who fits one of the above, who agrees to help on the condition that this character can travel with the group (e.g. the representative of a company who's funding an expedition or the son of a king who lent the group a ship).
Dryden Fassa in The Vision of Escaflowne is an Intrepid Merchant, with zero fighting skills, who manages to join Hitomi and Van's little gang by providing funding (read: an ungodly sum of money) for the eponymous mech's mid-season repairs.
Bubblegum Crisis: Sylia Stingray serves as both the leader and the backer for the Knight Sabers. She designs all of their equipment, including their hardsuits, and finances their operations, using her family's wealth, and by lining assignments with high paying officials, and businessmen.
In Dragon Ball Z, Bulma and her family fund and personally build the ludicrously expensive operations of building interstellar spacecraft, time machines, and the copious amounts of food that saiyans eat. Ox King and Hercule do too, to a lesser extent.
The Briefs family and Hercule are also the go-to people for hiding stuff that would compromise The Masquerade. When a bunch of Namekians find themselves stranded for nearly 9 months, Bulma invited them to stay at her place, and when they needed to keep Majin Buu under wraps for six months, Hercule's compound proved just the ticket. Capsule Corp. even seems to have a built-in Weirdness Censor due to the apparently copious oddities that Dr. Briefs' experiments have produced, so when a giant wish-granting dragon is summoned on his front lawn, nobody is particularly alarmed once they find out it's at Capsule Corp.
Angel acted in this capacity when he was with the Champions. He funded the group as well as being on the team. He also funded and led The Defenders, turning them into an actual team instead of a non-team.
On the X-Men he also sometimes provided funds, although at least at the beginning they mostly came from Charles Xavier, whose "old money" family fortune also included the Xavier Mansion and its spacious grounds and enough money to have his X-Men ride into action in Rolls-Royces from the first issue.
Nighthawk (the Marvel Comics version of Batman) provided funds for the Defenders while he was on the non-team, and they met at his ranch estate.
Iron Man's civilian identity, Tony Stark, originally funded The Avengers, setting them up in his mansion and giving them the use of his butler Jarvis.
This later was replaced by a foundation named after Stark's mother.
Other equipment, most notably the Quinjets, was produced in Wakanda and provided to the team by the Black Panther.
Kate Bishop (the Hawkeye of Young Avengers) provides funding for the team using her father's wealth, and converts one of his old buildings into a base for their operations. Unlike most examples, not only is she a fully fledged member of the group, she sometimes also acts as their unofficial leader.
R. J. Brande funded the Legion of Super-Heroes. In the current version (based mostly on the original one) he gained his fortune by making suns.
Reed Richards is this to the Fantastic Four, not only providing them with their technical equipment, but also with a big part of the funding through patent fees etc.
Ruben in Oceans Eleven. All he does is provide the initial investment and snark.
Inception: Saito, a ludicrously wealthy CEO, hires the team to perform the inception, and insists on going with them to protect his investment. He buys an entire airline to make it easier for them to get to their target, but isn't much use after that.
The Life Aquatic with Steve Zissou has the "bond company stooge" who (if I remember correctly) is an accountant who joins the crew to report back to their benefactors and make sure they stay within budget.
Villainous example; The Neimodian race from Star Wars is cowardly and lacks military expertise, but their participation in the Separatist Confederacy provides it with resources, since they control the Trade Federation. In the novellization of Revenge of the Sith, the Neimodian viceroy Nute Gunray tries to call General Grievous out, pointing that all of Grievous' sucess is funded by Neimodian money. Grievous is not impressed.
Fight Club: The narrator is an example of this, since he blackmails his boss at a large automobile company into funding Fight Club, and presumably later Project Mayhem. Of course, also being Tyler, the same person provides all the leadership and creative force behind those operations too, but he is not aware of that at the time.
Eduardo Saverin in The Social Network. He's the team member who repeatedly steps in with the cheque book before the money starts rolling in.
In Aliens Ripley is this from the perspective of the soldiers (since she's just a civilian who happened to have see what they've been sent in to investigate).
Toad from Kenneth Grahame's The Wind in the Willows starts out as this. He provides the caravan for their trip but doesn't really give much else to the team apart from that and his dangerous, endless desire for adventure. He becomes pretty resourceful past the halfway mark of the novel, though.
The protagonist (a journalist) of The Lost World is essentially on the trip to authenticate everything they find (since the professor has been ridiculed for years for his theories).
In Dracula Lord Arthur Godalming aids in funding the hunt for Dracula by providing transportation, lodging, and the like. While Abraham van Helsing helps with his knowledge of vampire lore.
In the Ciaphas Cain novel "Caves of Ice", Cain has to put up with an annoying tech priest. Initially because he has some knowledge of what might be down there (thanks to his interest in xenobiology) and after that because he has an auspex which is needed to see where the ambulls and orks they're hunting are.
Live Action Television
In Prison Break one character invokes this trope to keep himself useful by memorising and destroying a map the rest of the cast are following.
In season two of Nikita, Birkhoff provides the money, the home base, and is the team's invaluable Techno Wizard.
A BBC adaptation of The Lost World had the protagonist secure his place on the trip by having his employers help fund it.
Smallville: Oliver provides the money that keeps his team stocked with whatever they need, and rebuild the Watchtower when it, its windows, or the various technology in it is inevitably damaged every other week. In Season Ten with Tess Mercer's Heel-Face Turn, the team gains the financial backing of LuthorCorp as well.
In Mass Effect 2, Miranda sort of fits this, being your liason with The Illusive Man, who technically owns the Normandy SR-2 and supplies you with funds, equipment, and intelligence. TIM would be this directly but he never joins the squad.
In the metagame of World of Warcraft, healers were like this (they were hard to level up so there weren't many about, but they were needed to survive in most of the endgame instances). This lead to the stereotype of a Jerk Ass who played a priest so other players would have no choice but to put up with them.
In Dragon Age II, Varric uses his wide network of connections to find jobs for Hawke, keep the Coterie off Anders' back, forge papers proving Fenris owns his ex-master's manor, and keep Merrill out of trouble with the City Guard.
Also there's Setzer, owner of the only airship(s) in the world.
In the Broken Sword series, George serves as this at least once (being a patent lawyer, while his fellow investigator Nico is a pennyless journalist) in order to explain how they manage to travel to the various places around the world.