"Data frame" seems a necessary term if students are trying to search for answers on StackOverflow, mailing lists, or their favorite search engine; "tibble" won't turn up many relevant results.
Plus "data frame" is distinct—there's no confusion with tables, matrices, or arrays (and if there is with data.tables it's self-inflicted)—but not overly general (rectangular data, dataset). It's portable, both within R (S3 methods, docs, etc.) and outside (to Python, Spark, and Julia at least). It also works fine within the tidyverse: I tend to use data_frame and as_data_frame more than tibble and as_tibble. The print method for tibbles does start with A tibble: though, which may require further explanation.
One problem, though, is whether to use "data.frame", "DataFrame", "dataframe", or "data frame". The first is most directly derived from R, but doesn't make sense in a Python environment (.frame is not a method of data). In pandas, Spark, Julia, and Maple (apparently) they're called "DataFrames", though for general usage the camel case seems overly technical. SO data frame users decided to make [data-frame] and [data.frame] synonyms of [dataframe], but the tag info page uses "data frame" when talking about the concept independent of a language. I like the idea of "data frame" as a general concept and the local spelling in context, but the disparity does make searching a pain if a search engine doesn't realize they're all the same thing.
Edit: R's docs use "data frame" for the concept (as opposed to the function), as well; see ?data.frame:
Data Frames
Description
The function data.frame() creates data frames, tightly coupled collections of variables which share many of the properties of matrices and of lists, used as the fundamental data structure by most of R's modeling software.
(h/t @Frank)