Does the following choice principle have a name?
Every finite dimensional subspace of a vector space has a complement. Equivalently, every line inside a vector space has a complementary hyperplane.
How strong/weak is it compared to other choice principles? It certainly follows from the existence of a basis, and as such is a consequence of the axiom of choice.
Feel free to edit the Tags, I wasn't sure which tag is appropriate.
As a lower bound on the strength of this, by this answer of mine on MO, "every line has a complementary hyperplane" implies the axiom of choice for finite sets of bounded cardinality. More precisely, that answer shows that if $X$ is a set such that there exists an $\mathbb{F}_p$-hyperplane in the space $\mathbb{F}_p(X)$ of rational functions with elements of $X$ as variables for all primes $p\leq n$, then there is a choice function on the subsets of $X$ of cardinality $\leq n$.