Several of the answers gently suggested that I get off my duff and figure out it myself so below are my results. I think that sentiment generally goes against the grain of this site but if you want something done right, you might as well do.... :)
I modified option #1 to take advantage of @Ty suggestion to use StringBuilder.Length = 0 instead of the Remove method. This made the code of the two options more similar. The two differences are now whether the constructor for the StringBuilder is in or out of the loop and option #1 now uses the the Length method to clear the StringBuilder. Both options were set to run over an outputStrings array with 100,000 elements to make the garbage collector do some work.
A couple answers offered hints to look at the various PerfMon counters & such and use the results to pick an option. I did some research and ended up using the built-in Performance Explorer of the Visual Studio Team Systems Developer edition that I have at work. I found the second blog entry of a multipart series that explained how to set it up here. Basically, you wire up a unit test to point at the code you want to profile; go through a wizard & some configurations; and launch the unit test profiling. I enabled the .NET object allocation & lifetime metrics. The results of the profiling where difficult to format for this answer so I placed them at the end. If you copy and paste the text into Excel and massage them a bit, they'll be readable.
Option #1 is the most memory efficiency because it makes the garbage collector do a little less work and it allocates half the memory and instances to the StringBuilder object than Option #2. For everyday coding, picking option #2 is perfectly fine.
If you're still reading, I asked this question because Option #2 will make the memory leak detectors of an experience C/C++ developer go ballistic. A huge memory leak will occur if the StringBuilder instance is not released before being reassigned. Of course, we C# developers don't worry about such things (until they jump up and bite us). Thanks to all!!
ClassName Instances TotalBytesAllocated Gen0_InstancesCollected Gen0BytesCollected Gen1InstancesCollected Gen1BytesCollected
=======Option #1
System.Text.StringBuilder 100,001 2,000,020 100,016 2,000,320 2 40
System.String 301,020 32,587,168 201,147 11,165,268 3 246
System.Char[] 200,000 8,977,780 200,022 8,979,678 2 90
System.String[] 1 400,016 26 1,512 0 0
System.Int32 100,000 1,200,000 100,061 1,200,732 2 24
System.Object[] 100,000 2,000,000 100,070 2,004,092 2 40
======Option #2
System.Text.StringBuilder 200,000 4,000,000 200,011 4,000,220 4 80
System.String 401,018 37,587,036 301,127 16,164,318 3 214
System.Char[] 200,000 9,377,780 200,024 9,379,768 0 0
System.String[] 1 400,016 20 1,208 0 0
System.Int32 100,000 1,200,000 100,051 1,200,612 1 12
System.Object[] 100,000 2,000,000 100,058 2,003,004 1 20