An alternative to Todd-Coxeter enumeration is Knuth-Bendix rewriting. Coxeter groups have pretty nice rewriting systems. As @Qiaochu Yuan mentions, every element of a Coxeter group has an expression as a “reduced” word (which for Coxeter groups, means “of shortest length”). Unfortunately, elements have multiple expressions as reduced words, so you have to be a little careful listing them. While there are some pretty easy combinatorial ways to check which element you've got, you might appreciate a simple way of always getting the same reduced word. This is called a (reduced, confluent) rewriting system.
The one for $B_3$ is quite simple:(Editted to correct reduction ordering)
s_i \cdot s_i &\mapsto& 1 & 1 \leq i \leq 3 \\ %
s_3 \cdot s_1 &\mapsto& s_1 \cdot s_3 \\%
s_3 \cdot s_2 \cdot s_3 & \mapsto &s_2 \cdot s_3 \cdot s_2 \\%
s_2 \cdot s_1 \cdot s_2 \cdot s_1 &\mapsto&s_1 \cdot s_2 \cdot s_1 \cdot s_2 \\
s_3 \cdot s_2 \cdot s_1 \cdot s_3 &\mapsto& s_2 \cdot s_3 \cdot s_2 \cdot s_1 \\
s_3 \cdot s_2 \cdot s_1 \cdot s_2 \cdot s_3 \cdot s_2 &\mapsto&s_2 \cdot s_3 \cdot s_2 \cdot s_1 \cdot s_2 \cdot s_3
Each line indicates a rule. If you have some products of $s_i$, you keep blindly applying these rules until you can't anymore. Notice each rule has a direction. Only apply them forwards, never in reverse. In a presentation, you might have to use the rules in reverse, but in a rewriting system the rules only go one way. If you cannot apply any of the rules, then you are done, and you have a reduced word. In fact, amongst all reduced words for that element, you have the one that comes first alphabetically.
The rules themselves are pretty simple. Repeated generators go away. Alphabetize 31 to 13, since the presentation told you they were the same. Alphabetize 323 since $(s_2 s_3)^3$ can be rewritten as the “braid relation” $s_3 s_2 s_3 = s_2 s_3 s_2$, and we want the alphabetically first version. Similarly, converting $(s_1 s_2)^4$ into its braid relation and choosing the alphabetically first one, you get the fourth rule. The only slightly surprising rule is the last one, which is just needed to alphabetize a more complicated Braid relation.