The Temperley-Lieb algebra has the same generators as the $S_n$ group algebra, and the same commuting relations, but its other relations are different. A nice diagrammatic interpretation can be seen in the Wikipedia article.

Is there a standard analogue for other Weyl groups? Perhaps some specialization of the Hecke algebra?

There is a standard basis for the Temperley-Lieb algebra, which can be indexed by 321-avoiding permutations: pick a reduced word for $v$ in $S_n$'s generators, and interpret the generators inside T-L instead. This association is well-defined, because the 321-avoiders are also the "fully commutative" elements, meaning that each has only one reduced word up to commutation relations.

This subset of $S_n$ is also the set of "lambda-cominuscule" elements (meaning, the $T$-action on the Bruhat cell $BvB/B$ includes the dilation action), and is an ideal in left/right weak Bruhat order. In Weyl groups other than $S_n$ the lambda-cominuscule condition is strictly stronger than "fully commutative".

Assuming the first question, do the lambda-cominuscules, or some other subset of W, similarly give a basis?

Of course references would be most appreciated.