I have 3-month, single-entry visit visa for the UAE. However, I would like to have a multiple entry visa. Can I change the current visa and, if not, what can I do?
You cannot change the current visa, you can only extend it.
To get a multiple entry visa:
Someone has to apply from the UAE on your behalf. This is usually a tour operator, a hotel, a permanent resident (or citizen), or a legal entity such as a company or establishment.
If you have freehold property in the UAE, you can get a multiple stay visa.
If you are traveling via Emirates, certain nationalities can apply online for a tourist multiple entry visa from emirates.com (the facility is actually provided by VFS global and also available at dubaivisa.net).