My first question regarding your NEX-5N is whether you are using the LA-EA2 adapter that lets you use Alpha-mount lenses. There is a 55-210mm lens that's native to the e-mount system you have now, but the 70-300mm lens you mentioned is actually a different mount. Sony has two lens mount systems - the A-mount is for the larger-bodied mirror-based DSLR and SLT models which took over from Minolta's film cameras...this is a completely different lens mount from the NEX model you have. Then there's the E-mount, which is a newer mount developed for mirrorless compact cameras like yours. You can't use E-mount lenses on the A-mount cameras, but you can get adapters that let you use the A-mount lenses on the NEX/E-mount cameras.
The downside of the adapters is that you lose stabilization...and of course they're bigger and bulkier and heavier. If you wanted to stay with native lenses designed for your NEX camera, currently there aren't too many long-reach telephoto options - the 55-210mm lens that I think you have now is one of the longest - there is a much more high-end and expensive 70-200mm F4 lens which won't have any more reach, but will be faster and higher quality - but MUCH more expensive...and there's a 24-240mm lens with a little more reach, but not enough to really be noticeable. Honestly moving up to 300mm would be noticeable, but not drastic...you could better fill the frame with reasonably close birds - such as 20-40 feet away - but you still couldn't do much with birds that are 80-100 feet away or more.
You have one more option that's not too expensive, would still be reasonably light, and would even let you keep using the same 55-210mm lens you have now...AND would get you a pretty nice step up in reach. Years ago, when superzoom point-and-shoot cameras were popular and DSLRs were still rare and big, there were tele-extenders made that could screw onto the filter threads on the end of the zoom cameras, and magnify their focal length by 1.7x. The optics on some of these were quite good, and many mirrorless camera shooters have discovered that these same teleextenders can be screwed onto the end of some current lenses, and give you a reasonably good quality system with much more reach - your 55-210mm lens is a very good candidate for this, because it is a reasonably small, thin lens that has 49mm lens threads at the end - the teleextenders usually range from 49mm to 58mm threads, and with thin cheap step-up rings can be mounted right on the end of your lens. If you add a 1.7x extender to your 55-210mm lens, it becomes a 94 - 357mm lens. The long end of the lens will give you significantly more reach at 357mm to get you much closer to the birds...and the lens will still autofocus and have stabilization just as before. I'd highly recommend you consider this as a reasonably cheap option to get more reach without having to change anything else in your system. There are 2 good models of extender that work well on the 55-210mm Sony lens: One is the Sony DH1758 1.7x extender, and the other is the Olympus T-Con 1.7x extender. The Sony DH1758 can be hard to find used - they seem to usually go around $170-200 on the market today - the Olympus can run around $150-200, but should be more readily available used. The only thing you need to attach these to your lens is a 'step up ring'...which can be found at any camera shop or Amazon usually for under $10. For the DH1758 extender, you need a 49-58mm ring, and for the Olympus T-Con, you need the 49-55mm ring. The only word of warning is to watch out for some bait-and-switch sellers online, who will tell you they have the DH1758, but once you order it they'll call you to tell you it's out of stock but they can switch it out for an even better one - they end up sending you a cheap Chinese-made junk lens.
I currently shoot with a Sony A6000, having upgraded from my NEX-5N - but I've used the 55-210mm lens with the Sony DH1758 combo for many years, and it works great. Check out this gallery from my NEX-5N with the 55-210mm lens in my local wetlands - almost all the shots in this gallery were using the Sony DH1758 1.7x extender on the camera and lens:
http://www.pbase.com/zackiedawg/nex5n_wakogc&page=all
Here's a photo of what the DH1758 looks like attached to the 55-210mm lens on the NEX-5N camera:
http://g1.img-dpreview.com/A572058D6C634B15BD121E0A738A914D.jpg
Good luck, and hope that helps!