There’s a great video by Peter Williams discussing names, and disambiguation of names, in the Gospels. Although he doesn’t address this example specifically, some of my thoughts are drawn from his.
Simon was the most common Jewish boy’s name in the time and place where Jesus lived. Unsurprisingly, there are a lot of Simons in the New Testament, which is why the narrators in the Gospels & Acts frequently disambiguate which Simon they’re talking about (e.g. Simon Peter, Simon the Zealot, Simon the leper, Simon the tanner, etc.)
I’ve worked in multiple office environments where there were several people named John, including the top boss. I noticed a pattern. When people were talking with one of the Johns, they just called him John. When they were talking about a John, the top boss was just “John”, whereas all the other Johns were “John X” or something like that. The most prominent individual with the name was the one who needed no introduction, all the other Johns were disambiguated from him, not he from them.
This would explain why the narrators frequently disambiguate among people with common names, even though the quoted dialogue sometimes does not. If two people were speaking to each other they didn’t need to specify which Simon, which John, which James, etc. But the narrator telling the story does because we weren’t there to see who is talking to whom.
The prominence concept is also one of the reasons the James of the Epistle of James is usually held to be James the Lord’s brother. After the death of James the son of Zebedee (circa 44), James the Lord’s brother is frequently referred to as just “James”, suggesting he was the most prominent Christian man of the time bearing that name. He was the James who needed no introduction, hence no need to disambiguate himself from other people named James when writing his epistle.
Not the companion of Cleopas
I agree with Tony Chan’s comments that the companion of Cleopas—while he could have been named Simon (most popular name after all)—is not the Simon we’re looking for. Cleopas and his companion met the 11, which includes 2 Simons, so neither of those Simons is Cleopas’ companion. If Cleopas and his companion are the ones talking in verse 34 (which I believe is a rather strained reading of the passage), they should say “the Lord appeared to us”.
The sense of the passage is it’s the group they joined—the eleven and those with them—who are speaking when “the Lord appeared to Simon” is stated.
Let’s apply the thoughts above and see if we can narrow down the possibilities. There’s no disambiguation of “Simon” in this passage, suggesting either:
A. There was one and only one person named Simon present during the conversation, and so it was obvious which Simon was meant.
B. The speaker pointed to one of the Simons to indicate which Simon was meant
C. The Simon referenced is the most prominent Simon, therefore no disambiguation needed.
We can rule out A since both Simon Peter and Simon the zealot were present. B is technically possible but unhelpful. I suggest C is the most likely possibility, both because Jesus had made/would make Simon Peter a leader among his brethren, and because it is explicitly attested by Paul when he lists appearances of the resurrected Jesus:
And that he was seen of Cephas, then of the twelve: (1 Cor. 15:5)
Cephas saw the Lord before the others did (Further discussion of this passage on this site here). Cephas is used several times in the New Testament to refer to Peter. Cephas comes from the Aramaic “Kepha” and Peter from the Greek “Petros”; both refer to the man we know as Simon Peter (and both are a form of the word "rock").
To me the statement by Paul seals the deal for option C—“Simon” in the OP’s question is a reference to Simon Peter, and there was no disambiguation not only because Simon Peter was present, but also because he was the most prominent Simon.