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You might investigate Singular, a software package for algebraic polynomial computations. I know little about it, but it does implement a so-called Hilbert-driven Buchberger algorithm, which (somehow!) finds "an appropriately chosen fast" ordering of the monomials, specifically to circumvent the problem that "the performance of Buchberger's algorithm is sensitive to the choice of monomial order." Their documentation provides one example with a $100 {\times}$ speedup.

This article by Manuel Kauers in Scholarpedia may help. Here are some quotes:

Change of Ordering

Some applications require Gröbner bases with respect to a particular ordering of the power products for which Buchberger's algorithm is not as efficient as for other orderings. In such situations it may be advantageous to first compute a Gröbner basis with respect to some ordering where Buchberger's algorithm runs faster and in a second step transform this Gröbner basis to a Gröbner basis for the desired ordering.

Gröbner Walk

Two different techniques for performing such a change of ordering are known. One is known as Gröbner walk. It is based on an interpretation of orderings as regions in a space. If two orderings correspond to regions which overlap, then a Gröbner basis for one of the orderings can be turned into a Gröbner basis for the other by calling Buchberger's algorithm on a small auxiliary problem for which it usually terminates quickly. When the regions for two orderings do not overlap, it is always possible to connect them by a path consisting of orderings where the regions of any two consecutive ones have an overlap. The transformation can then be done step by step, switching in each step to the next ordering on the path. [...]

Linear Algebra

The second technique uses linear algebra. If $G$ is a Gröbner basis for some ordering, then we have [...] Using this technique, [...], one can determine the elements of a Gröbner basis with respect to an ordering different from the ordering of $G$.

See the article for more details and references.

2 Added second reference and quotes.
This article by Manuel Kauers in Scholarpedia may help. Here are some quotes:

Change of Ordering

Some applications require Gröbner bases with respect to a particular ordering of the power products for which Buchberger's algorithm is not as efficient as for other orderings. In such situations it may be advantageous to first compute a Gröbner basis with respect to some ordering where Buchberger's algorithm runs faster and in a second step transform this Gröbner basis to a Gröbner basis for the desired ordering.

Gröbner Walk

Two different techniques for performing such a change of ordering are known. One is known as Gröbner walk. It is based on an interpretation of orderings as regions in a space. If two orderings correspond to regions which overlap, then a Gröbner basis for one of the orderings can be turned into a Gröbner basis for the other by calling Buchberger's algorithm on a small auxiliary problem for which it usually terminates quickly. When the regions for two orderings do not overlap, it is always possible to connect them by a path consisting of orderings where the regions of any two consecutive ones have an overlap. The transformation can then be done step by step, switching in each step to the next ordering on the path. [...]

Linear Algebra

The second technique uses linear algebra. If $G$ is a Gröbner basis for some ordering, then we have [...] Using this technique, [...], one can determine the elements of a Gröbner basis with respect to an ordering different from the ordering of $G$.

See the article for more details and references.

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You might investigate Singular, a software package for algebraic polynomial computations. I know little about it, but it does implement a so-called Hilbert-driven Buchberger algorithm, which (somehow!) finds "an appropriately chosen fast" ordering of the monomials, specifically to circumvent the problem that "the performance of Buchberger's algorithm is sensitive to the choice of monomial order." Their documentation provides one example with a $100 {\times}$ speedup.