Increment base-36 strings

Japt, 13 bytes

n36 Ä s36 ù0A

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Takes input as a string

Explanation

n36            converts input to base 36
    Ä           +1
      s36       to base 36 string
          ù0A   left-pad with 0 to length 10

JavaScript (ES6), 45 bytes

Saved 4 bytes thanks to @O.O.Balance

s=>(parseInt(1+s,36)+1).toString(36).slice(1)

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Haskell, 58 bytes

d=['0'..'9']
f s=snd(span(<s)$mapM(\_->d++['a'..'z'])d)!!1

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A very brute-force strategy: generate all the length-10 base-36 strings in order, and find the one that comes after the input in the list. Take an enormous amount of time on strings far from the start of the list.

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Haskell, 60 bytes

q '9'='a'
q c=succ c
f(h:t)|any(<'z')t=h:f t|r<-'0'<$t=q h:r

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Reads the string left to right until it reaches a character followed by a suffix of all z's, which may be empty. Increments that character, and replaces the z's with 0's.