In my country of residence there's a possibility to deduct charities from the tax. Since most have a daytime job and the tax is deducted in advance, in practice it means that at the end of the year one would get some money back from the state by having less tax deducted from the next salary. If someone makes a neder of making a donation of a given amount, has (s)he fulfilled the obligation if a certain part is reimbursed?
R Shimon Taub (The Laws of Tzedakah and Maaser, p. 149) writes that
One who benefits by giving tzedakah because he ends up paying less taxes is not obligated to give that profit to tzedakah. Therefore if one would have had to pay $10,000 in taxes had he not given tzedakah, but because of all the tzedakah he gave must pay only $7,000, he is not obliged to give in the profit of $3,000 to tzedakah.
He references Igrot Moshe YD 1:143 that even real benefit coming from tzedakah belongs to the donor and not to tzedakah. As such one would have fulfilled one's obligation even if reimbursed in part.
An interesting point raised in Rav Moshe’s responsum is the deduction of charitable contributions when computing income tax. Certain governments allow a tax deduction for donating money (or even goods) to charities, which lowers the tax bill. Are we required to give maaser based on the higher amount of post-tax net income which results from taking the tax deduction (since we pay less tax as a result of the tax deduction, we are left with a higher net income), or do we compute the maaser based on what would have been a lower amount of net income, assuming no tax deduction? Rav Moshe rules that we need only give the maaser on the lower amount of the posttax net income, and can take the benefit from the tax deduction allowed for charitable donations (Igros Moshe, Yoreh Deah, vol. I, 143).
and I have now seen it quoted in the name of R Chaim Kanievsky as well (here).
To give an alternate perspective, Eretz Hemda (very bottom here) suggests giving 10% of the "tax-back" as further tzedakah if the person has the means to do this (remember there is no formal set of halachot for tzedaka which many consider a worthwhile custom but not an obligation, which explains why different people sometimes come to different rulings in this area)
You could look at it as income. You can also look at it as it turning out that you paid less taxes than was expected during withholding, and therefore that retroactively you had more net income in the previous year than you thought and thus did not give enough maaser (which is fine).
Either way of analyzing it, it is proper to give 10% of the refund in the next "maaser period." If you have difficulty giving a full maaser, there is room to look for various leniencies and you can be back in touch with us.
Finally note it is best not to make vows (nedarim) to charity lest one forget to honor his commitment and violate a Torah prohibition. Saying bli neder after a pledge is an easy way to accomplish this (see R Avrohom Chaim Feuer's The tzedakah treasury, p. 220)