Gaia, 25 bytes

ṇ+†ṅ⟨)₌¤:~+4ṛ⟨ṛ₌¤⟩ₓ⟩¦$

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Thanks to Kevin Cruijssen for pointing out 2 bugs!

ṇ				| delete the last character from the input
 +†				| push the input again and concatenate together, so for instance
				| 'abc' 'bc' becomes ['ab' 'bc' 'c']
 ṅ				| delete the last element
 ⟨ 		⟩¦	| for each of the elements, do:
	)₌			| take the first character and push again
	 ¤			| swap
	 :			| dup
	 ~			| swap case
	 +			| concatenate strings
	 4ṛ		| select a random integer from [1..5]
	 ⟨ ⟩ₓ		| and repeat that many times
		 ṛ₌¤		| select a random character from the string
		 	| clean up stack
			 $	| convert to string

Note that 4ṛ is because ṛ is implemented for an integer z as python's random.randint(1,z+1), which returns an integer N such that 1<=N<=z+1.

APL (dzaima/APL), 23 bytes

Anonymous tacit prefix function.

∊2(⊣,?4⍴⍨?5⊇,,-⍤,)/

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2(…)/ apply the following infix tacit function between each character pair:

 - the switchcase

 ⍤ of

 , the concatenation of the pair

 ,, prepend the concatenation of the pair to that

 …⊇ pick the following elements from that:

  ?5 random number in range 1…5

  4⍴⍨ that many fours

  ? random indices for those

∊ ϵnlist (flatten)

Perl 6, 60 bytes

S:g.)>(.)=$/~[~] roll ^5 .roll+1,$/.lc,$/.uc,$0.lc,$0.uc

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The lowercase/uppercase part is kinda annoying.

Jelly, 12 bytes

;Œsṗ5X¤XṭṖµƝ

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Bash, 121 bytes

-20 bytes thanks to Nahuel

-9 bytes thanks to roblogic

for((i=0;i<$#1;i++))
s=$1:i:1
m=$1:i:2
m=$m,,$m^^
for((t=0;t++<RANDOM%6;))
s+=$m:RANDOM%4:1

printf "$s"

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Original answer

Bash, 150 bytes

Have done very little golf bashing and trying to improve my bash, so any comments welcome.

for((i=0;i<$#1-1;i++));do
c=$1:$i:1
n=$1:$((i+1)):1
a=($n $c, $c^ $n, $n^)
shuf -e $a[@] -n "$(shuf -i 1-5 -n 1)"|xargs printf %s
done

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Code is straightforward loop through chars setting current c and next n character, then creating an array of the 4 possibilities, repeating one of them so there's exactly 5. Next we shuffle that array, and then choose n elements from it, where n itself is random between 1 and 5.

Python 2, 107 bytes

f=lambda s:s and s[0]+''.join(sample((s[:2]+s[:2].swapcase())*5,randint(1,5)))+f(s[1:])
from random import*

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05AB1E, 18 17 bytes

ü)vyн5LΩFyD.š«Ω]J

Inspired by @Giuseppe's Gaia answer.

-1 byte thanks to @Shaggy.

Try it online 10 times or verify all test cases 10 times.

Explanation:

ü) # Create all pairs of the (implicit) input
 # i.e. "Hello" → [["H","e"],["e","l"],["l","l"],["l","o"]]
 v # Loop over each these pairs `y`:
 yн # Push the first character of pair `y`
 5LΩ # Get a random integer in the range [1,5]
 F # Inner loop that many times:
 y # Push pair `y`
 D.š« # Duplicate it, swap the cases of the letters, and merge it with `y`
 Ω # Then pop and push a random character from this list of four
 ]J # After both loops: join the entire stack together to a single string
 # (which is output implicitly as result)

Charcoal, 27 bytes

ＦＬθ«Ｆ∧ι⊕‽⁵‽⭆✂θ⊖ι⊕ι¹⁺↥λ↧λ§θι

Try it online! Link is to verbose version of code. Explanation:

ＦＬθ«

Loop over all of the indices of the input string.

Ｆ∧ι⊕‽⁵

Except for the first index, loop over a random number from 1 to 5 inclusive...

‽⭆✂θ⊖ι⊕ι¹⁺↥λ↧λ

... extract the previous and next characters from the string, take the upper and lower case versions, and pick a random character of the four.

§θι

Print the character at the current index.

perl 5 (-p), 77 bytes

s/(.)(?=(.))/$x=$1;'$x.=substr"U$1$2L$1$2",4*rand,1;'x(1+5*rand)/gee;s/.$//

TIO

Japt -P, 14 bytes

äÈ+Zu pv ö5ö Ä

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äÈ+Zu pv ö5ö Ä :Implicit input of string
ä :Take each consectutive pair of characters
 È :Pass them through the following function as Z
 + : Append to the first character of the pair
 Zu : Uppercase Z
 p : Append
 v : Lowercase
 ö : Get X random characters, where X is
 5ö : Random number in the range [0,5)
 Ä : Plus 1
 :Implicitly join and output

Python 3, 167 bytes

from random import*;c=choice
def f(s):
 i=0;r=""
 for i in range(len(s)-1):
 r+=s[i]
 for j in range(randrange(5)):r+=c([str.upper,str.lower])(c(s[i:i+2]))
 return r

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Jelly, 14 bytes

;;;Œs$Xɗ¥5X¤¡Ɲ

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Explanation

 Ɲ | For each overlapping pair of letters
; | Join the first letter to...
 5X¤¡ | Between 1 and 5 repetitions of...
 Xɗ¥ | A randomly selected character from...
 ;;Œs$ | A list of the two letters and the swapped case versions of both

C(GCC) 175 162 bytes

-12 bytes from LambdaBeta

f(s,S,i,r,a)char*s,*S,*i;srand(time(0));for(i=S;*(s+1);++s)a>64&a<91?a^32:a:a)a=rand()&1?*s:*(s+1);*i=0;

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Scala 2.12.8, 214 bytes

Golfed version:

val r=scala.util.Random;println(readLine.toList.sliding(2).flatMapcase a :: b :: Nil=>(a +: (0 to r.nextInt(5)).map_=>((c: Char)=>if(r.nextBoolean)c.toUpper else c.toLower)(if(r.nextBoolean)a else b)).mkString)

Golfed with newlines and indents:

val r=scala.util.Random
println(readLine.toList.sliding(2).flatMap
 case a :: b :: Nil=>
 (a +: (0 to r.nextInt(5)).map_=>
 ((c: Char)=>if(r.nextBoolean)c.toUpper else c.toLower)(if(r.nextBoolean)a else b)
 )
.mkString)

Ungolfed:

import scala.io.StdIn
import scala.util.Random

def gobble(input: String): String = 
 input.toList.sliding(2).flatMap 
 case thisChar :: nextChar :: Nil =>
 val numberOfAdditions = Random.nextInt(5)
 (thisChar +: (0 to numberOfAdditions).map _ =>
 val char = if(Random.nextBoolean) thisChar else nextChar
 val cc = if(Random.nextBoolean) char.toUpper else char.ToLower
 cc
 )
 .mkString


println(gobble(StdIn.readLine()))

Perl 5 -n, 61 bytes

s/.(?=(.))/print$&,map(maplc,uc$&,$1)[rand 4]0..rand 5/ge

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C# (Visual C# Interactive Compiler), 236 213 209 bytes

a=>int i=0,j;var m=new Random();var s="";var c = a.Select(x=>Char.IsLetter(x)?(char)(x^32):x).ToArray();for(;i<a.Length-1;i++)for(j=m.Next(1,5);j-->0;)s+=new[]a[i],c[i],a[i+1],c[i+1][m.Next(0,3)];return s;

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T-SQL query, 286 bytes

DECLARE @ char(999)='String'

SELECT @=stuff(@,n+2,0,s)FROM(SELECT
top 999*,substring(lower(c)+upper(c),abs(v%4)+1,1)s
FROM(SELECT*,number n,substring(@,number+1,2)c,cast(newid()as varbinary)v
FROM(values(1),(2),(3),(4),(5))F(h),spt_values)D
WHERE'P'=type and n<len(@)-1and h>v%3+2ORDER
BY-n)E
PRINT LEFT(@,len(@)-1)

Try it online unfortunately the online version always show the same result for the same varchar, unlike MS SQL Server Management Studio

C# (Visual C# Interactive Compiler), 156 bytes

n=>var k=new Random();int i=0;foreach(var j in n.Skip(1).Zip(n,(a,b)=>a+b))for(Write(j[1]),i=0;i++<k.Next(5);)Write(k.Next()%2<1?j.ToUpper():j.ToLower());

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Japt -P, 43 16 bytes

äÈ+(Zv +Zu)ö5ö Ä

Shortened by a lot now!

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C (gcc), 110 bytes

i,p;g(char*_)for(i=rand()%1024,putchar(*_);p=_[i%2],putchar(i&2&&p>64&~-p%32<26?p^32:p),i/=4;);_[2]&&g(_+1);

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i,p;g(char*_)
 for(i=rand()%1024, //initialize: get 10 random bits 
 putchar(*_); // and print current char
 p=_[i%2], //1st bit => current/next char
 putchar(i&2&& //2nd bit => toggle case
 p>64&~-p%32<26 // but only if char-to-print is alphabetic
 ?p^32:p),
 i/=4;); //discard two bits; run at least once
 _[2]&&g(_+1); //if not last char, repeat with next char

The number of characters printed (per input character) is not uniformly random:

1 if i< 4 ( 4/1024 = 1/256)
2 if 4<=i< 16 ( 12/1024 = 3/256)
3 if 16<=i< 64 ( 48/1024 = 3/ 64)
4 if 64<=i< 256 (192/1024 = 3/ 16)
5 if 256<=i<1023 (768/1024 = 3/ 4)