What does DIM stand for in Visual Basic and BASIC?

It's short for Dimension, as it was originally used in BASIC to specify the size of arrays.

DIM — (short for dimension) define the size of arrays

Ref: http://en.wikipedia.org/wiki/Dartmouth_BASIC

A part of the original BASIC compiler source code, where it would jump when finding a DIM command, in which you can clearly see the original intention for the keyword:

DIM    LDA XR01             BACK OFF OBJECT POINTER
       SUB N3
       STA RX01
       LDA L        2       GET VARIABLE TO BE DIMENSIONED
       STA 3
       LDA S        3
       CAB N36              CHECK FOR $ ARRAY
       BRU *+7              NOT $
       ...

Ref: http://dtss.dartmouth.edu/scans/BASIC/BASIC%20Compiler.pdf

Later on it came to be used to declare all kinds of variables, when the possibility to specify the type for variables was added in more recent implementations.

Variable declaration. Initially, it was short for "dimension", which is not a term that is used in programming (outside of this specific keyword) to any significant degree.

http://in.answers.yahoo.com/question/index?qid=20090310095555AANmiAZ

Back in the day DIM reserved memory for the array and when memory was limited you had to be careful how you used it. I once wrote (in 1981) a BASIC program on TRS-80 Model III with 48Kb RAM. It wouldn't run on a similar machine with 16Kb RAM until I decreased the array size by changing the DIM statement