What was given to Cara Dune at the end of The Mandalorian S2E4?

It’s Captain Teva’s own Alliance badge.

He most likely gave it to her as a show of respect for her loss as well as subtle invitation to her that states, “I respect you… You are always welcome to reconsider the offer…”

According to Slash film; bold emphasis is mine:

“This episode brings back Paul Sun-Hyung Lee as Captain Carson Teva and makes me wonder if the New Republic will have a larger role in future episodes and seasons. He talks to Cara Dune about her background as an Alderaanian, which is a touching moment, but also makes me wonder who the hell asks someone whose planet was destroyed if they ‘lost anyone.’ Of course she did. After intruding on her pain and apologizing for her loss, he gives her a service medal with a Rebel symbol on it and walks away. It’s unclear if this was his service medal to honor his time on Alderaan, or something else, but if that’s the case, it would make for a touching moment.”

And here is what Wookiepedia says about the exchange; again bold emphasis is mine:

“After Dune silently denied his appeal, he then pointed out that she was from Alderaan, asking who she had lost during the planet's destruction. The marshal claimed she had lost everyone during the disaster, and the captain gave his condolences before leaving his Alliance badge with her and going back to his X-wing.”

Nothing solid past that at this point. But based on some of the cynicism expressed by Captain Teva and Trapper Wolf in the second episode of season two (“The Passenger”) — as well as Cara’s own dismissal of being invited to be a part of the New Republic — my guess would be that Captain Teva gave it to her as an acknowledgment of her own personal pain in the context of everything including his relatively minimal loss compared to hers.

Meaning, in this ABY (“After the Battle of Yavin”) world, a medal is just a medal. The Rebellion won the, but the war to keep order in the universe — as well as hunt down rogue pockets of Imperial hold-outs — still goes on. Why does he need that badge if she’s suffered worse than him?

It could also be a way of him saying, “Look… You are upset and sorry about reopening old wounds… This door is always open if you want to reconsider…”

UPDATE: As of the 6th episode of the 2nd season — aka: “Chapter 14: The Tragedy” — we learn that Cara Dune has now become a Marshal of the New Republic:

“There, Djarin learns that Carasynthia Dune has become a Marshal of the New Republic, saying that he has heard rumors that she has gone legit.”

So one can assume that object was a New Republic Marshal’s badge and after some thought, she accepted Captain Teva’s badge and is now one of the few, the proud — and clearly overworked — Marshal’s attempting to keep order in the galaxy.

While the idea that it's a medal makes a lot of sense, another possible explanation that also fits the show's "space western" vibes is that it's the Star Wars equivalent of a "sheriff's badge". Note that he is depicted as a sort of cop on the beat, or a detective, and Cara herself is the town's Marshal.

Perhaps his gesture is a way of saying that she's doing his job better than he; or an offer to keep doing what she does not simply as someone, but as a representative of the New Republic.

edit: welp, looks like I guessed right =)

It gets explained in Season 2 Episode 6 - The Mandolorian visits Cara and is holding the badge:

Cara Dune, Marshal of the New Republic

So, Captain Teva was basically investing as a marshal of the law for Navarro (or, at least, the town and its surrounds)