I am not a certified electronics engineer, so I bow to those who are. Having said that, I don't think there is anything nasty about this.
These article explains how an inverting operational amplifier circuit works.
http://www.national.com/an/AN/AN-31.pdf
or
http://www.swarthmore.edu/NatSci/echeeve1/Ref/SingleSupply/SingleSupply.html
and
or
As you can probably see. The circuit you are looking for is the first, basic inverting amplifier.
Essentially, the gain is the ratio of voltages across the input and output. That is set with two resistors.
The golden rules of Op Amp behaviour with feedback (from "The Art of Electronics") are:
1 The output attempts to do whatever is necessary to make the voltage difference between the two inputs (+ and -) zero.
2. The inputs draw no current.
So if the + input goes to 1V, and the ratio of the two resistors is 1:2, then the output will go to 2V.
Negative feedback is a way to set the gain. You just need a gain of roughly 2. Further, the badwidth is very low, it's only the rate that you can twiddle the potentiometer, so almost any op amp should work.
I think you'd like an op-amp that works at 5V, and ideally are capable of rail-to-rail voltage swings (so that you can get as much amplification as practical).
The paper at swathmore recommends TLCV2772 or TLC2774 because they are rail-to-rail.
I had a look at Farnell, I found about two dozen on the UK site:
The Art of Electronics (TAoE) recommends using a pair of resistors with a reasonable value to keep the output impedance reasonably low, I.e. the sum of the two resistors in this case would be about 10K-100K. The Arduino analogue inputs likes to see in output impedance of the previous stage (the output of the op amp) of 10K or less. On the other hand, you'd like the two resistors to be about 10x bigger than the input potentiometer, so that they don't load it.
If you are okay with something a bit less fancy, TAoE recommends TLC27M2A (or related op amps).
At Farnell the TLC27M2A is very cheap http://uk.farnell.com/jsp/search/browse.jsp?N=0&Ns=PRICE_PLS_006_PRICE1|0&Ntk=gensearch_001&Ntt=TLC27M2A&Ntx=mode+matchallpartial
and starts as cheap as 0.31 GBP.
HTH
GB