The Lubavicher rebbe had a practice of giving out dollar bills to people. If I were to have one originally given to someone else would it be permissible to sell such a memento?
The Rebbe gave the dollars to people to give to charity. People usually substituted it for another dollar, to keep the one from the Rebbe. Nevertheless, if one did give it to charity, of course that's alright.
In this case, the dollar was given to you. If it was given to you to keep; then halachikly they'd be no problem, unless it was given to you on condition you don't sell it. It's no different halachikly then any random dollar.
I, as a chossid, wouldn't sell it, being that it's from a Rebbe, so it's holy and special. Giving one to a child, or someone who needs a blessing... is a different story, but I wouldn't sell it. From your question, I assume you probably don't feel that way, though I recommend keeping it.
See ליקוטי שיחות חלק ט"ז פרשת בא (ה), where it speaks about not selling something from a Rebbe to a gentile or making it hefker. Whether here, where the dollar was given to give to tzedaka, where the money can end up anywhere, is different, I can't really answer that.
If the dollar is not officially yours, then no, you cannot. Selling it would be equivalent to selling someone else's property. Furthermore, it's money, so I don't know what that means halachically (if I sell a stolen dollar for $2, and normally the stolen property would have to be paid back in double...).
That said, if it is your property, then yes you could sell it or do whatever you like with it. It is, after all, a dollar.
What might be more valuable to you however is to give it away. The purpose of the Rebbe giving dollars out was as charity, and not to those who needed the money (mostly). Money was only the tool to share the blessing. Because that's all money is, a tool. So you could use it to, say, purchase pork to eat and boost negativity, or you could purchase a kosher meal and boost positivity.
No matter what you do with it, the dollar remains a tool, and the blessing remains with it. My suggestion is that if you have no use for it yourself, that you give it to charity and allow that blessing to become stronger through your own gift. Or, if you have a desire to grow, spend the dollar towards that. That in turn will put the dollar into someone else's hands and grow as well, along with the rest of the money spent (what does a dollar buy these days?). There's no right or wrong answer with what you can do with the dollar; ultimately all money you spend will work the same way, as a tool. Only you decide whether that helps or hinders your spiritual growth.