While I think you could do with refining your question with a bit more detail (how much of the line has to be within to qualify, how do you want to handle lines coincident with polygon edges, etc... I do have a thought on generally how you might be able to approach this problem. Yes, this would be a relatively elaboarate method to accomplish what should be pretty easy to do, but... it should work none-the less. One way would be to build a model or script that would use a GP tool to split all of your line segments at the polygon boundary (I would probably do this with clip tool using a layer with selection applied for the polygon to clip by so it only uses the polygon(s) in question. If you have the output of that tool go to a geodatabase and you have your initial dataset in a geodatabase, then you would just have to do a table to table join on the two line files (original and output of GP tool) and then calculate the percentage difference between the two Shape.Len fields (if a line segment in original file is 100 ft long, and that same line segment, clipped to the polygon, is 75ft long, then 75% of that line would be within the polygon). Then you could just do a select by attribute based on this percent.
Again, I know this is rather involved, it creates an intermediary dataset, and there probably is an easier way to do this that I don't know about; but, if you write a model or script to do such, and delete the output line FC when you're done with it, it shouldn't appear all that difficult to the end user... even if it's a bit of a pain to build initially.
Just some thoughts, and let me know if that doesn't make sense and I'll try and explain better.