Better random (unique) file name

It might be best to use CreateUUID.

CreateUUID[]

 "73ccc27c-687f-4eca-8214-ceeb8a8b7773"

The Properties & Relations section shows a way to express this string as an integer if that's what you're after:

FromDigits[StringReplace[CreateUUID[], "-" -> ""], 16]

296740835687065620982102887154699649600

This should give you strings that are unique. It uses $SessionID and $ProcessID which are a unique combination by definition, either hashed (almost unique) or plain (unique by design).

Short name, almost unique.

Hash in "Base36String" to keep the string short (28 characters). It has the date and $KernelID in plain text for easy identification.

StringJoin[
 Riffle[
  {
   Hash[{$SessionID , $ProcessID}, "CRC32", "Base36String"],
   DateString[{"Year", "MonthNameShort", "Day", "Hour24", "Minute", 
     "Second", "MillisecondShort"}],
   ToString[$KernelID]
   }, "-"]
 ]

"02y4q0o-2019Jul04180113431-0"

The probability of collision for a "CRC32" hash is extremely low, $(n - 1) / 2^{32}$ and way better than your $(n - 1) / 10^5$, and occurring at the same time (within a millisecond) and in the same kernel is in practical terms imposible.

---

Long names, absolutely unique

If you can afford extremely long names, you could leave $SessionID and $ProcessID unhashed.

You can shorten the string length also using IntegerString with "Base64" encoding.

StringJoin[
 Riffle[
  Flatten@{
    StringDelete[
     IntegerString[{$SessionID , $ProcessID}, "Base64"], {"+", "/", 
      "="}],
    DateString[{"Year", "MonthNameShort", "Day", "Hour24", "Minute", 
      "Second", "MillisecondShort"}],
    ToString[$KernelID]
    }, "-"]
 ]

Otherwise use them to define folder (directory) names.

FileNameJoin[
 ToString /@ {
   $MachineName,
   $SessionID ,
   $ProcessID,
   DateString[
    {"Year",
     "MonthNameShort",
     "Day",
     "Hour24",
     "Minute",
     "Second",
     "MillisecondShort"
     }]
   }
 ]