There are a lot of members here who can help, but some more information would be nice. I did a Google search for "lava lamp with color-changing leds" and
the results were a lot of inexpensive lava lamps. So it may not be economically sane to try to reverse-engineer your LED controller.
OTOH, this IS a hobby site, so cost may not be your first concern. One solution would be to design and populate a brand new circuit board that does what you want: slow down the color rotation. I would add to such a PCB the capability to control (with a potentiometer or push-button switches) the rate at which the colors change and perhaps a control to adjust the intensity of the colors. All that would be required is a small Microchip PIC processor and "some software" plus the PIC programming tools available at the Microchip website. The cost of the PIC microprocessor is "down in the noise" when it comes to budget. The biggest cost is the time required to design and build the replacement LED controller.
Depending on your skill-set and relevant experience, it could take anywhere from a few weeks to several months to create a working prototype. You might be able to finish this before the holiday season begins if you intend the lava lamp project to be a gift. Members of this forum can help, but you would be responsible for all the "grunt work" of purchasing components and making a circuit board. It doesn't have to be a manufactured printed circuit board, although those are very inexpensive. You would have to learn PCB layout techniques and possibly download PCB design software, such as the free KIcad program, to have a printed circuit board (PCB) manufactured.
I don't recommend it, but you could reverse engineer your existing LED controller, removing all the components and visually tracing out the circuits those components are connected to. If that allows you to identify the microprocessor, and if it is a common microprocessor that the hobbyist can "play" with, then it might be possible to simply replace the microprocessor with a new one that you have programmed. This is a LOT of work with no guarantee of success. Which is why I would recommend that you make it a project here on maker.pro to develop a new LED driver that does what you want it to do.