I think that using colnames and rownames makes the most sense; here's why.
Using names has several disadvantages. You have to remember that it means "column names", and it only works with data frame, so you'll need to call colnames whenever you use matrices. By calling colnames, you only have to remember one function. Finally, if you look at the code for colnames, you will see that it calls names in the case of a data frame anyway, so the output is identical.
rownames and row.names return the same values for data frame and matrices; the only difference that I have spotted is that where there aren't any names, rownames will print "NULL" (as does colnames), but row.names returns it invisibly. Since there isn't much to choose between the two functions, rownames wins on the grounds of aesthetics, since it pairs more prettily withcolnames. (Also, for the lazy programmer, you save a character of typing.)