Lucas proved a congruence for binomial coefficients mod a prime $p$ that uses the base $p$ digits of the two numbers in the binomial coefficient. See http://en.wikipedia.org/wiki/Lucas%27_theorem. It was extended to multinomial coefficients by Dickson.
Stickelberger's congruence for Gauss sums (not to be confused with Stickelberger's congruence for the discriminant of a polynomial or number field) involves products of factorials of base $p$ digits. It is discussed in Section 11.3 of Lemmermeyer's "Reciprocity Laws from Euler to Eisenstein" and in chapter 1 of Lang's book on cyclotomic fields. The congruence lifts $p$-adically to the Gross--Koblitz formula for Gauss sums.
Addition of numbers, expressed in terms of base expansions, is an elementary school example of cohomology. More precisely, the carry-digit function is a cocycle. See http://www.math.wayne.edu/~isaksen/Expository/carrying.pdf.
Dirichlet's theorem about primes in arithmetic progression could be interpreted in terms of digits in special cases, e.g., 1/4 of all primes have last decimal digit 1, 3, 7, or 9 (Dirichlet for modulus 10), or there is an equal proportion of primes having any string of two digits $ab$ not divisible by 2 or 5 as its last two digits (Dirichlet's theorem for modulus 100). In the other direction, with leading digits, the situation is less satisfactory: the set of primes with leading decimal digit 1 does not have a natural density within the set of all primes.
You should be cautious about pursuing "digit theory"within number theory too far, since it doesn't have a good reputation, the results of Lucas, Dickson, and Stickelberger notwithstanding. For instance, there is a review on MathSciNet about a paper involving digits that ends with the following remark: "There is also a list of serious number theory papers, by Lucas, Kummer, and others, that mention digits (usually to a prime base). But the reviewer is not convinced thereby that Smith numbers are not a rathole down which valuable mathematical effort is being poured." Smith numbers, Keith numbers, and emirps (that is not a typo) are all part of number theory in a broad sense, but not in a mainstream professional sense. Appearances can sometimes be deceiving: automorphic numbers may at first look totally recreational, but they are connected with the Chinese remainder theorem, Hensel's lemma, and the contraction mapping theorem. In a different direction, the statistical properties of digits in a fixed base for irrational numbers or continued fraction entries for irrational numbers have profound connections to ergodic theory, and if tomorrow someone proved $\pi$ is a normal number or that the continued fraction entries of $\sqrt[3]{2}$ are unbounded (Lang and Trotter did some computer calculations on that -- see an appendix to a recent edition of Lang's "Introduction to Diophantine Approximations") it would be an amazing development, probably as much as if someone showed the fractional parts of $(3/2)^n$ for $n = 1,2,3,...$ are equidistributed in $[0,1]$. It's not clear from the question if you considered "digit theory" to include such aspects of positional representations that are not directly about positive integer digits.
