I also like Andre's term ``quasi-categories'' and prefer it to the infinity alternatives, for the reasons he gives. Like Kan complex, it has a fixed and evocative definite meaning, unlikely to be confused with anything else.
The term "(infinity,1)-category" is not so much meant as an alternative for "quasi-category", but as a intentionally less specific term that subsumes concepts that are different from, but equivalent to, quasi-categories. Such as Kan-complex-enriched categories or complete Segal spaces, or algebraic quasi-categories, or categories with weak equivalences, or... When doing abstract higher category theory it is useful to be able to speak, for instance, of the (infinity,1)-category of all small infinity-groupoids and its abstract properties, without having to specifically fix a concrete model in terms of which this entity may be brought to paper. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]