Your wife is right, of course. My Russian is pretty rusty, I’m afraid.
We can always try to eliminate the actual Slavic languages one by one. Here’s Russian:
Добрый день, Григорий. Приходи со (за*) мной
Dobry den’, Grigory. Prikhodi so (za*) mnoy.
Good day, Gregory. Come with me.
*I think “za” is what a Russian would use for this, but I am not a fluent speaker.
Ukranian for “Good day” is Добрий день, very similar to Russian. Byleorussian is Добры дзень. The Slovene word for day is dán. Czech is den. Slovak is also den, but with a mark over the n, and Polish is dzien, with another mark over the n.
I had a crazy idea that the vowel <u> in “dobru” and “Grigoru” may be meant to mimic the shape of Cyrillic <и> (the vowel /i/). Russian chat alphabet or Volapjuk spelling use roman ASCII letters to mimic Cyrillic letters that way. Which raises the question of why it also uses a regular Roman <i>. Maybe because both graphemes exist in Ukrainian? I’ve also seen Roman <y> used for /u/.
I typed in a sentence in romanized Korean and the guesser guessed “French.” When I typed it in using the Korean alphabet, the guesser guessed “Unknown.”
I typed “I am a black man” and it said Welsh. (By the way, I am Caucasian.)
It’s absolutely not Romanian. It’s absolutely Slavic. 100%.
I guess given the further background by Wendell, I’m guessing it’s just a Slavic hodgepodge made-up for the novel, to give the character a sense of Eastern European-ness, without pinning him/her down to a specific country.
In Russian there’s no matching pronoun that declines to “min”. The verb “prihodit’” does not make sense in the imperative with the preposition “so”.
Closest Russian sentence would be:
Dobriy den, Grigoriy. Prihodi so mnoi.
And it’s almost non-sensical unless perfectly contextualized as an answer to the question
“A s kem mne prihodit’?” (And with whom am I supposed to come?)
Literally it would be like “Good day, Grigoriy. Come with ME along!” (Only makes sense in English as an answer to a question using ‘along’, otherwise it would be just ‘Come with me!’)
This is also non-sensical in Ukrainian and Belarusian as far as I can tell.
Now enough with what it ain’t and let’s figure out what it am:
I’m stumped. As far as I can tell “со мин” (“so min”) is not a valid preposition-pronoun combination in any Slavic language. Nor is “Prihud so” or any other variants. I’m thinking it’s a very very contorted version of something scribbled in Cyrillics on a napkin, possibly originally in Macedonian 
Anthony Boucher is long dead, so contacting him is not an option.
Something just occured to me.
Whatever the language, Grigoru is the Slavic name for Gregory. What if instead of a novel, it showed up in Boucher’s radio show The Casebook of Gregory Hood. That would make a lot of sense, because it would be easy to make a couple of spelling mistakes while hearing the language, and in any case, Anthony could well have gotten the phrase by asking a friend of his who maybe studied Russian or whatever for a semester, which would further goof the phrase up. Kind of a stretch, but it would make sense.
I typed in some Vietnamese (chúng ta nęn cám ơn) and that page thinks it’s Irish_utf8, whatever that is.
Heh - I just type in ‘you are a fool’ in flawless Japanese - it said it was Malay.
Slacker.
I gave it a sentence in my conlang Mömö - Mü meni Mömö gelki ba the anun tonïmoyath - and it said “Hungarian,” a reasonable guess since Hungarian is one-fourth of the source material.
I gave it a sentence in Uzbek and it said “Turkish.” Not too far off, but it obviously could stand a bit of fine tuning. It must not have all the national languages of the world programmed into it, let alone minority ones. So it would probably mistake an obscure Slavic language for a more prominent one. Anybody know Sorbian, Kashubian, or Old Church Slavonic?
There exists a pan-Slavic conlang as pulykamell speculated. Panie i panowie, meet Slovio!
Bulgarian, perhaps? I’ll check with my household Bulgarian speaker when he comes in.
The language guesser thought Hawaiian was Swahili, which is pretty off considering Hawaiian doesn’t have T, B, J, D, Z, V, or S, and Swahili does.
A shame, though-- that tool could be useful in my line of work.