The edit or Levenshtein distance between two strings is the minimum number of single symbol insertions, deletions and substitutions to transform one string into another. For example $$\operatorname{E}(01010,00100)=2.$$
Let $E_n$ be a random variable giving the edit distance between two random binary strings of length $n$.
What bounds can be found for
$$\lim_{n \to \infty} \frac{\mathbb{E}(E_n)}{n} = c\;?$$
According to my non-extensive simulations, $c \approx 0.288$.
There is related work by Luecker on the expected length of the longest common subsequence which has lower and upper bounds of $0.788071n$ and $0.826280n$. This was a line of work originally started by Chvatal and Sankoff. However the longest common subsequence length and $n$ minus the edit distance need not be similar.
Best Answer
The only rigorous bound I am aware of is due to Gonzalo Navarro*
$$c\geq 1-{\rm e}/\sqrt{\sigma},$$
for an alphabet of $\sigma$ characters. Obviously, for the binary string ($\sigma=2$) this bound is ineffective. Navarro also mentions a large-$\sigma$ conjecture $c=1-1/\sqrt{\sigma}$, which for the binary string would give $c=0.2929$, quite close to your numerical finding.
*G. Navarro, A guided tour to approximate string matching (2001)