I would first like to discuss the concept of lateralization and clear up a common misconception about hemispheric dominance.
"A brain is considered to be asymmetrical (or lateralized) if one side (hemisphere or other brain region) is structurally different from the other and/or performs a different set of functions." (Bisazza et al., 1998).
A good example is the lateralization of language which relates to handedness. It has been widely accepted that a majority of people are left dominant for language, although right dominance increases with the degree of left handedness (Knecht et al., 2000).
Lateralization and behavioural asymmetries are neither restricted to language or humans (Bisazza et al., 1998). In the scientific community, hemispheric dominance is usually discussed in the context of a specific function. However, there is a common misconception by those who have an incomplete understanding of brain lateralization, particularly worsened by the spread of popular science articles and pseudoscientific tests (such as the one linked to in the comments) that wrongly assumes there is an overall dominance of one side of the brain (with left corresponding to logical and analytical thinking and right corresponding to emotional and creative thinking), that determines your personality trends. This is an incorrect oversimplification that is explained in a good article by RationalWiki.
Thus, your question is misguided in focusing on changes in hemispheric dominance. Even if you focused on change of lateralization for a specific function, this is unlikely as it requires substantial reorganization of the brain that is really only observed during development and to a much smaller extent following brain injury. However it is still good to question what changes there may be due to using a non-dominant hand more often! In fact (Teixeira and Teixeira, 2007) looks at how practice with the non-preferred hand alters manual preference for a specific task, and suggests this generalizes to other related motor tasks (Teixeira and Okazaki, 2007). The task used in their 2007 study involves sequentially touching the thumb with (in order) the ring, middle, little, and index fingers. The goal is to complete 5 cycles without interruption in the shortest time possible. They assess this before, after, and 30 days following practice, observing a noticeable shift in those that are allowed to practice compared to controls. You can choose a similar task and/or something different (for example, score in a quick video game that requires pointing with the trackball) and measure how you improve in comparison to a friend who doesn't practice with the left hand.
Ironically, they discuss how the left premotor cortex is dominant for learning sequential tasks (whether performed with right or left hand) and so might limit the observable shifts. For this reason it might be good to try their task to compare with that study and also your own task for personal interest. If you are really interested in how performance and practice (especially long term) can affect the brain, there is lots of existing literature and imaging studies on the differences in brains of musicians compared to non-musicians or amateurs. Due to years of practice, professional musicians often have differences in and sometimes larger motor areas (Schlaug, 2006), increased cerebellar volume (Hutchinson et al., 2003), increased auditory cortical representation (Pantev et al., 1998) and increased assymetries such as in the planum temporale (Schlaug et al., 1995).
