When I look for drop-in replacements for MOSFETs, I aim for:
* Same package (TO-220)
* Same pinout (usually same package = same pinout, but there are occasional surprises...worth double-checking just to be sure)
* Same or higher maximum Vds (>= 30V in this case)
* Same or higher maximum Vgs (>= +/- 20V here, luckily this is pretty standard)
* Similar Vgs(th) turn-on thresholds: 1-3V range for 150N03 vs. 2-4V range for IRF3305 seems pretty reasonable - as long as you don't use one with a 5-7V range to replace one with a 1-3V range.
* Same or lower Rds-on: the 150N03 has a max. 4 mOhms @ Vgs=10V (pay attention to the conditions, because there's a difference between the "typical" and "max." specs, and also the Rds-on changes with Vgs), while the IRF part has a max. 8 mOhms @ Vgs=10V...I'm a bit confused why someone would suggest this as a drop-in replacement because the resistance is twice as high as the original. It's possible that conduction losses (heating due to the current through this on-resistance) aren't the dominant factor here, but that's hard to know on somebody else's design without a lot of testing, etc.
* Similar capacitances: look at the Ciss and Coss vs. voltage curves when available (like figure 5 in the IRF3305 datasheet), unfortunately the 150N03 is a massively under-specified part with a terrible datasheet, which doesn't include these. However, you can at least compare the single Ciss and Coss points it gives in the electrical characteristics table, at Vds=25V, to see that they're pretty comparable to the IRF3305's same numbers. These are more important the higher the switching frequency gets, but in this case I'd guess that the motor control PWM is comparatively slow (maybe a few kHz), so as long as your replacement MOSFET doesn't have twice the capacitance it's probably fine.
The MOSFET part numbers I know off the top of my head are all for seriously-higher-frequency applications, but check Digikey: their site's undergoing maintenance right now so I can't post a link, but if you look in the "Discrete Semiconductors - MOSFETs, Single" category there's a nice parametric search where you can narrow down the big parameters you want (like Rds-on and package type), then sort by price and start looking at datasheets.