Suppose there is a bill to increase the tax on cigarettes by $1 per pack coupled with an income tax cut of $500. Suppose a person smokes an average of 500 packs of cigarettes per year—and would thus face a tax increase of about $500 per year from the cigarette tax at the person’s current level of consumption. The income tax measure would increase the person’s after-tax income by $500. Would the combined measures be likely to have any effect on the person’s consumption of cigarettes? Why or why not?

First, the answer to this question is subjective. Those measures might cause a person to smoke less, but those combined measures might also cause a person to smoke more. Both can be equally defended.
You could also defend a null hypothesis, meaning that no change would occur in the packs consumed per year. This is what I would personally defend. 500 packs per year is a lot of smoking. That's 1.4 packs per day. The average consumption is 1 pack, so the smoker in this example is already considered a heavier smoker than normal. In my opinion, the cost (or savings difference) is likely so negligible that a person's level of cigarette consumption probably wouldn't change.
What I also don't know is what this hypothetical person's annual income is in the first place. Additionally, I don't know what their current debt situation is. If his/her car and house are both paid off, and they make $250,000 per year, I don't think a $500 change either way is going to matter.
Furthermore, it is conceivably possible that this person stops smoking cigarettes cold turkey for reasons other than the cost of keeping the habit or the savings from quitting the habit. Smoking is detrimental to health. The person might scale back from 500 packs per year to zero packs per year in order to hopefully live longer or avoid the various cancers that accompany cigarette usage.
http://healthsurvey.hscic.gov.uk/data-visualisation/data-visualisation/explore-the-trends/smoking/adult/median-smoking.aspx

Many peer-reviewed studies have found that increasing sales taxes on cigarettes decreases smoking among both children and adults, particularly young adults (see the link below from Tobacco Free Kids and the source below). For example, a 2012 report by the Surgeon General called "Preventing Tobacco Use Among Youth and Young Adults" stated that a 10% increase in cigarette prices results in a 3%–5% decrease in the consumption of cigarettes. It is true, however, that adults are more resistant to price increases in cigarettes than younger people are, so if this person is older, they may be less likely to give up smoking. In addition, younger people in particular are tuning to vaping nicotine, so if this person is a younger person, they may start vaping instead of smoking.
In addition, according to the link below from Market Watch, more than 44% of Americans do not pay income taxes, as their take-home pay is too low, or they are not working. Therefore, a tax reduction may not affect them.
Source:
HHS, Preventing Tobacco Use Among Youth and Young Adults: A Report of the Surgeon General, 2012, http://www.surgeongeneral.gov/library/reports/preventing-youth-tobacco-use/full-report.pdf.
https://www.marketwatch.com/story/81-million-americans-wont-pay-any-federal-income-taxes-this-year-heres-why-2018-04-16

https://www.tobaccofreekids.org/assets/factsheets/0146.pdf

There is a problem in missing assumptions of the question, namely that the effect of changes in cigarette prices affects all demographics equally. That actually isn't the case. In general, cigarette taxes reduce smoking in the young and the economically disadvantaged, but have little effect on middle-aged smokers in higher economic brackets. Thus this answer needs to be parsed in terms of demographics.
For people with middle class incomes, the effect is likely to be negligible, with the decision of whether to continue smoking determined by other issues such as the availability of smoking cessation programs and personal choice. According to the CDC, people below the poverty level are twice as likely to be smokers as those living above the poverty line. Education is also a crucial factor, with only 3 percent of people with postgraduate degrees smoking but 24 percent of high school dropouts smoking. Thus, even though taxation is unlikely to change the smoking behavior of the well-educated and affluent, that population is already unlikely to smoke.
For people living in poverty, the income tax cut would be irrelevant, as over 80 percent of people with incomes under $50,000 have no tax liabilities. Since the income tax cut would have no effect on them, the increased cigarette prices would result either in reduced smoking or increased black market cigarette purchases.
https://money.cnn.com/2013/08/29/pf/taxes/who-doesnt-pay-federal-income-taxes/

https://www.cdc.gov/tobacco/data_statistics/fact_sheets/adult_data/cig_smoking/