Don't make it an easily searchable name, as far as i know private streams don't show up on a search (I use the term "private" loosely since site mods can see even those)
Adding stuff to "don't make an easily searchable name", I wouldn't attempt this. a 20-character stream that is named "klgslfixoekdjfkdkciw" would certainly violate spam policies. I'd rather see someone discover it by guessing it.
Sooo, basically you want to leech imgflip's bandwidth so you can have your own private online gallery to hotlink memes or photos from without having to re-upload them every time you want to post them.