In addition to JPEG2000 (.jp2) you could also consider ECW (now owned by ERDAS). It is also lossy like the JPEG method but can produce very good results with relatively fast opening times. Another option to consider would be MrSID by LizardTech. MrSID uses lossless wavelet compression and in some cases MrSID even performs better than JP2 despite being lossless!
While both ECW and MrSID have free readers both for proprietary and open source (GDAL) GIS tech, neither has a free writer (I don't think), though both are available through ArcGIS (from memory). I have noticed compression artifacts using both ECW and JP2, so the theoretical compression achievavble may well not be practical depending on the sensitivity of your data.
With a tif image itself you can commonly use LZW compression, which is a lossless compression method. In addition to LZW, TIF images can also be compressed using JPEG and PackBits and many GIS sytems will give you an option, though often default to LZW (probably because it is a safe option being lossless). Though since JP2 tends to give better results than plain jpeg and the other two results, then this may not give you sufficient compression for your needs. However, it is not clear from your post whether the tif has any compression initially, so perhaps the LZW might be sufficient and could be worth considering as it is no longer a proprietary method and is lossless.
Basically, there are tons of methods depending on what tech you can afford, who is going to be consuming the imagery (i.e. can they open it and use the data) and how much you need to compress the data (i.e. you maybe you must have a massive compression ration, or maybe something more modest will suit your use-case). I would hazard a guess that your tiff might already be compressed, as this could explain your increase in RAM requirements since the compressed image must be uncompressed in memory and many systems automatically apply LZW to Tifs. ArcGIS defaults to LZ77 as its compression of choice (which is a lossless method like zip and is what a PNG uses) so that might be the case here (just guessing - I forget whether Arc applies lossless compression automatically to tifs or not, though I have a feeling it does). It will be worth checking whether there is any compression already on your tif before you choose another method as you may find the comparative additional compression disappointing otherwise.
As you are using ArcGIS have a look at the documentation where it lists the compression methods commonly available.