For fish tissue samples, freezing is best (avoid preserving in formalin/ethanol if possible).  cost varies quite a bit among labs, but for bulk 13C/15N, you should be able to analyze fish samples for ~ $10/sample.  In terms of your question about accuracy, isotopes are complementary to stomach contents.  Whether they are more/less useful really depends on the circumstance and specific questions you are asking.  

As Carlos has already pointed out, the question of which approach to use depends on the questions you wish to get answers to.  Gut content tells you what your eels have been feeding on prior to capture (prior being a function of gastric emptying rate).  Stable isotope data can give you a diet chronology if you like. How far back in time this information will go depends on growth or turn-over rate of the specific tissue you are analysing. I have caried out analyses of fish otoliths for collegues in Scandivia for this purpose (13C for trophic level; 18O for water temperature). Another source for information on trophic level is 15N abundance in proteinous tissue (e.g. collagen; keratin; muscle). As Carlos said the stable isotope lab at UC Davis should be able to analyse samples for you (and give advide on sample collection & storage).  Other stable isotope labs in the US I would recommend are: USGS, Reston (Tyler Coplen) and University of Utah (Jim Ehleringer).

Dieter - Very accurate. Excellent for determining trophic level and for determingin source of carbon (allochthonous vs. autochthonous). We have used isotopes in Lake Erie for Silver Chub and other species.  Feel free to call if you want to discuss.

Aside from the excellent suggestions mentioned above, it is worth mentioning that you will likely need to take samples of the possible food sources of the eels, and/or samples of an isotopic "baseline" (the producer level) as well.

For high lipid content tissues such as eel flesh a correction might be needed because lipids are depleted in 13C relative to proteins and carbohydrates.

This is an excellent review of the use of stable isotopes in food webs: 

Layman, C. A. et al. (2012), Applying stable isotopes to examine food-web structure: an overview of analytical tools. Biological Reviews, 87: 545–562. doi: 10.1111/j.1469-185X.2011.00208.x

study d15N is used to infer the trophic position of an organism, whereas d13C indicates mainly the food source. you can obtain both isotopic signals for 15 Euros per sample, and you can find technical support in University of Toulouse (France).

I hope my answer helped you.

Biochemical components, able to give information, significantly related to geographical origin, life history and nutritional status of fish are fatty acids. You can decide to monitor also these parameters in addition to isotopes. In some studies, fatty acids can represent a very good alternative.

We've been involved in numerous food web studies over the years, both terrestrial and aquatic.  Take a look at our web site and feel free to contact me with any questions.

My colleagues and I have used stable isotopes to establish trophic relationships among fishes in a highly contaminated river-estuary.   Yellow eels, as expected fell relatively "high" in the food web based on their known feeding preferences, and 15N muscle tissue content.  I have attached the relevant paper and supplement. 

It looks as if others have contributed helpful information here.  I also want to point out that the utility of stable isotopes for studying diet can be limited by whether there is isotopic variability among potential prey items, which is necessary to be able to distinguish differences in diet if they exist.  If you are trying to make comparisons within populations, individuals could have diets that are composed of different prey items that are isotopically similar.  Thus, using only a stable isotope approach would not help you in that situation.  But if you want more general trophic level and habitat information, the isotopic approach can be quite useful.

Yes, but in our particular case, and for much of our work, prey signatures weren't critical.  It was the primary producers, highly zoned by C3 and C4 pathways, by salinity and water depth that were the drivers of the signatures in our fish. Regards,

Mike