Integrate-and-fire neuron models are the workhorse for most studies in computational neuroscience. They describe the evolution of the membrane potential on a domain with fixed reset and threshold values. Whenever the membrane potential exceeds the threshold value, an action potential is generated, this is the integrate-and-fire mechanism. Both the timing and the rate of these action potentials are of fundamental importance for the understanding of neural coding. For a theoretical description, typically one or more stochastic differential equations are used to model spike times. However, many experimental studies suggest that the spiking threshold in vivo is highly variable and e. g. depends on the firing rate. Spike threshold variability has been described phenomenologically in modified Hodgkin-Huxley models to explain for example threshold fatigue. We here describe a theoretical framework for minimal noisy threshold models, and show how the mean first passage time may depend non-monotonically on the noise strength.