Aims: We observed a new cataclysmic variable (CV) SDSS J080434.20+510349.2 to study the origin of long-term variability found in its light curve. Methods: Multi-longitude, time-resolved, photometric observations were acquired to analyze this uncommon behavior, which has been found in two newly discovered CVs. Results: This study of SDSS J080434.20+510349.2 concerns primarily the understanding of the nature of the observed, double-humped, light curve and its relation to a cyclic brightening that occurs during quiescence. The observations were obtained early in 2007, when the object was at about V˜17.1, about 0.4 mag brighter than the pre-outburst magnitude. The light curve shows a sinusoidal variability with an amplitude of about 0.07 mag and a periodicity of 42.48 min, which is half of the orbital period of the system. We observed in addition two “mini-outbursts” of the system of up to 0.6 mag, which have a duration of about 4 days each. The “mini-outburst” has a symmetric profile and is repeated in approximately every 32 days. Subsequent monitoring of the system shows a cyclical behavior of such “mini-outbursts” with a similar recurrence period. The origin of the double-humped light curve and the periodic brightening is discussed in the light of the evolutionary state of SDSS J080434.20+510349.2. Table [see full textsee full textsee full text] and Fig. [see full textsee full textsee full text] are only available in electronic form at http://www.aanda.org Individual photometric measurements are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/486/505